Meta tags:
description=A blog about probability and olympiads by Dominic Yeo;

Headings (most frequently used words):

bmo1, convex, ordering, and, probability, post, navigation, eventually, almost, everywhere, finitely, many, solutions, 2023, why, do, we, need, the, lebesgue, integral, borel, cantelli, lemmas, bmo2, 2022, 2021, on, galton, watson, trees, hoeffding, inequality, 2019, blog, about, olympiads, by, dominic, yeo, 1a, extension, problems, main, menu, aside, recent, posts, previous, articles, categories, blogroll, stats, spam, blocked,

Text of the page (most frequently used words):
the (805), that (305), and (297), this (256), for (190), are (182), which (102), with (99), have (94), but (87), one (82), #problem (78), not (72), can (66), you (64), all (58), more (57), then (54), some (53), there (49), when (47), square (45), from (45), many (44), two (44), would (42), even (42), these (42), number (41), since (41), than (41), will (40), convex (39), where (39), random (38), integers (38), note (37), first (37), functions (37), olympiad (36), about (35), set (35), only (34), probability (33), such (33), may (30), angle (30), question (30), sequence (30), problems (29), equal (29), example (28), here (28), now (27), function (27), other (26), numbers (26), odd (26), also (26), has (26), negative (26), prime (25), inequality (24), ordering (24), any (24), angles (24), case (24), argument (24), simple (23), theory (23), sum (23), integral (22), without (22), prove (22), because (22), non (22), consider (22), given (22), limit (21), general (21), post (21), might (21), values (21), bmo1 (20), what (20), how (20), possible (20), particular (20), infinitely (20), geometry (19), posted (19), like (19), whether (19), construction (19), useful (18), they (18), well (18), each (18), show (18), true (18), must (18), could (18), replacement (18), over (18), mean (18), write (17), time (17), distribution (17), british (17), measure (17), point (17), most (17), solutions (16), however (16), let (16), same (16), second (16), above (16), setting (16), condition (16), answer (16), permutation (15), mathematical (15), 2022 (15), need (15), finitely (15), reply (15), good (15), know (15), just (15), three (15), fact (15), while (15), statement (15), was (15), defined (15), find (15), holds (15), original (15), finite (15), almost (14), large (14), hoeffding (14), often (14), definition (14), much (14), clear (14), positive (14), course (14), sampling (14), distributions (14), taking (14), factors (14), eventually (13), 2012 (13), lebesgue (13), out (13), been (13), using (13), indeed (13), always (13), does (13), trying (13), very (13), between (13), natural (13), follows (13), total (13), examples (13), cyclic (13), uniform (12), process (12), theorem (12), galton (12), watson (12), january (12), 2013 (12), why (12), lines (12), triangle (12), get (12), think (12), integer (12), down (12), doesn (12), work (12), them (12), study (12), students (12), sums (12), define (12), say (12), proof (12), four (12), into (12), required (11), blog (11), offspring (11), euclidean (11), equations (11), november (11), 2014 (11), previous (11), solution (11), right (11), infinite (11), ways (11), least (11), after (11), element (11), weight (11), directly (11), result (11), means (11), make (11), measurable (11), isosceles (11), stochastic (10), coupling (10), part (10), 2017 (10), borel (10), tagged (10), figure (10), both (10), proving (10), really (10), chosen (10), end (10), certainly (10), final (10), see (10), multiple (10), context (10), few (10), dominicyeo (10), bound (10), form (10), obtain (10), either (10), zero (10), way (10), use (10), different (10), level (10), main (10), independent (10), lemma (10), conclusion (10), variables (9), bmo (9), december (9), 2016 (9), 2018 (9), common (9), important (9), less (9), check (9), studying (9), should (9), paper (9), following (9), though (9), discussed (9), gives (9), events (9), wrong (9), start (8), size (8), iid (8), october (8), trees (8), algebra (8), 2015 (8), 2019 (8), 2021 (8), leave (8), turn (8), had (8), apply (8), terms (8), meet (8), never (8), choice (8), isn (8), something (8), step (8), value (8), equation (8), under (8), found (8), before (8), particularly (8), notion (8), sometimes (8), shown (8), immediately (8), our (8), small (8), already (8), thinking (8), powers (8), standard (8), below (8), integrals (8), your (7), comment (7), convergence (7), tree (7), markov (7), property (7), walk (7), generating (7), new (7), maths (7), mathematics (7), april (7), june (7), 2023 (7), extension (7), cantelli (7), include (7), probably (7), key (7), except (7), alternative (7), line (7), term (7), rather (7), every (7), helpful (7), otherwise (7), solve (7), coefficients (7), involving (7), significant (7), moment (7), concentration (7), exactly (7), setup (7), monotone (7), earlier (7), still (7), elements (7), configuration (7), once (7), instead (7), meaning (7), various (7), results (7), increments (7), require (7), suppose (7), years (7), smaller (7), avoid (7), pieces (7), games (7), sets (7), alice (7), graph (6), february (6), march (6), bmo2 (6), lemmas (6), diagram (6), via (6), direct (6), its (6), better (6), construct (6), actually (6), cannot (6), showing (6), satisfies (6), aren (6), enough (6), primes (6), steps (6), round (6), weighted (6), order (6), method (6), measures (6), version (6), bounds (6), certain (6), works (6), establish (6), likely (6), their (6), greater (6), essentially (6), exercise (6), involves (6), notation (6), leading (6), clearly (6), approach (6), perfect (6), interesting (6), impossible (6), factor (6), take (6), try (6), triangles (6), chasing (6), too (6), equivalent (6), structure (6), including (6), format (6), worth (6), relatively (6), easy (6), range (6), close (6), bc2 (6), reasonable (6), spaces (6), sigma (6), faults (6), everywhere (5), variable (5), combinatorics (5), chebyshev (5), branching (5), surely (5), inequalities (5), diophantine (5), july (5), august (5), september (5), search (5), tangent (5), idea (5), being (5), copies (5), red (5), don (5), anything (5), through (5), points (5), lies (5), necessary (5), induction (5), exact (5), assert (5), next (5), extend (5), quadratic (5), appear (5), cases (5), full (5), best (5), risk (5), increasing (5), class (5), ho63 (5), bounded (5), bps18 (5), probabilities (5), pairs (5), want (5), return (5), later (5), assume (5), corresponding (5), whenever (5), nothing (5), bit (5), lhs (5), uniformly (5), won (5), list (5), height (5), population (5), exists (5), group (5), considering (5), mind (5), hard (5), consecutive (5), larger (5), opposite (5), left (5), similarly (5), parity (5), similar (5), valid (5), obvious (5), vectors (5), split (5), ask (5), squares (5), circumcircle (5), draw (5), bc1 (5), coloured (5), green (5), union (5), containment (5), nearest (5), rounding (5), gon (5), differ (5), bob (5), create (4), comments (4), biased (4), law (4), arithmetic (4), local (4), exchangeable (4), cambridge (4), sure (4), times (4), processes (4), discrete (4), interest (4), linear (4), recent (4), posts (4), video (4), did (4), didn (4), generally (4), extra (4), degrees (4), especially (4), success (4), characterisation (4), perpendicular (4), respectively (4), option (4), exist (4), starting (4), possibilities (4), perhaps (4), leads (4), ten (4), five (4), divisible (4), open (4), look (4), hope (4), copyright (4), permission (4), successive (4), azuma (4), optimal (4), references (4), authors (4), short (4), event (4), called (4), formally (4), known (4), contexts (4), generated (4), vertices (4), handle (4), saying (4), definitely (4), based (4), usual (4), sufficient (4), comparison (4), convexity (4), sense (4), applications (4), referred (4), valued (4), symmetric (4), rhs (4), difference (4), around (4), those (4), equally (4), goal (4), situation (4), control (4), crucial (4), future (4), says (4), observation (4), relations (4), nonetheless (4), far (4), recently (4), readers (4), complete (4), retained (4), proceed (4), count (4), achieve (4), aim (4), ideas (4), writing (4), counting (4), twenty (4), nice (4), relevant (4), thoughts (4), reading (4), seems (4), along (4), pair (4), difficulty (4), focus (4), third (4), side (4), year (4), colour (4), slightly (4), whose (4), statements (4), takes (4), approximating (4), riemann (4), specifically (4), reduction (4), kite (4), axz (4), sixth (4), website (3), wordpress (3), com (3), content (3), sign (3), view (3), reader (3), supercritical (3), domination (3), sandwich (3), coalescence (3), martingale (3), deviation (3), brownian (3), teaching (3), functional (3), analysis (3), 2024 (3), relation (3), added (3), lots (3), within (3), apb (3), 180 (3), read (3), off (3), lie (3), circle (3), doing (3), choosing (3), calculation (3), dominic (3), ensure (3), wording (3), length (3), consideration (3), six (3), latter (3), precisely (3), reason (3), fashion (3), who (3), sat (3), brief (3), commentaries (3), official (3), until (3), jensen (3), loss (3), rearrangement (3), chernoff (3), application (3), arxiv (3), bm05 (3), online (3), samples (3), comparing (3), again (3), sample (3), making (3), threshold (3), worse (3), seen (3), addition (3), longer (3), model (3), consistent (3), concentrated (3), complicated (3), generate (3), observations (3), sort (3), plenty (3), possibility (3), harder (3), variance (3), initially (3), shortly (3), remains (3), quantities (3), suffices (3), wouldn (3), versions (3), article (3), further (3), strassen (3), provide (3), during (3), hold (3), rvs (3), convenient (3), denote (3), satisfy (3), attack (3), independently (3), recall (3), serte (3), meeting (3), across (3), classical (3), related (3), thought (3), clue (3), introduce (3), bad (3), lead (3), adding (3), arguments (3), external (3), quadrilateral (3), equality (3), diagrams (3), couple (3), unless (3), equilateral (3), lengths (3), practice (3), solving (3), straightforward (3), competition (3), discussion (3), aside (3), excellent (3), quite (3), initial (3), allowed (3), potentially (3), seeing (3), forced (3), personally (3), able (3), long (3), arun (3), give (3), constraints (3), took (3), spent (3), written (3), exhaustive (3), described (3), reproducing (3), advised (3), normally (3), yourself (3), past (3), successful (3), subsets (3), vector (3), dimensional (3), constant (3), polya (3), handled (3), undergraduate (3), types (3), another (3), polynomial (3), sandwiching (3), coefficient (3), coming (3), quadrilaterals (3), place (3), lecturing (3), handful (3), monkey (3), pathologies (3), permutations (3), complements (3), proofs (3), uncontroversial (3), countable (3), algebras (3), anyway (3), depending (3), issues (3), supremum (3), improper (3), defining (3), pattern (3), dots (3), follow (3), accurate (3), discuss (3), objects (3), density (3), cubes (3), free (2), own (2), email (2), log (2), spanning (2), summer (2), school (2), strong (2), stopping (2), statistics (2), sparse (2), power (2), poisson (2), mixing (2), levy (2), principle (2), international (2), imo (2), component (2), exploration (2), excursion (2), central (2), motion (2), blocked (2), spam (2), annoying (2), university (2), exchangeability (2), walks (2), martingales (2), articles (2), older (2), navigation (2), conjecture (2), recurrence (2), circles (2), imagine (2), evidence (2), several (2), twice (2), awkward (2), call (2), outside (2), stuck (2), namely (2), satisfying (2), essential (2), crucially (2), options (2), pick (2), couldn (2), neither (2), 200 (2), remove (2), understand (2), collection (2), thing (2), cycles (2), period (2), whereas (2), last (2), digit (2), half (2), looking (2), topic (2), questions (2), challenging (2), held (2), stop (2), averse (2), oh69 (2), orders (2), az67 (2), explicit (2), link (2), conditional (2), weights (2), special (2), becomes (2), decreasing (2), compare (2), remaining (2), proportion (2), procedure (2), motivation (2), investigating (2), exploring (2), plus (2), additional (2), multiset (2), includes (2), removing (2), repetitions (2), stated (2), distinct (2), expectations (2), cross (2), existence (2), viewed (2), gain (2), preferred (2), ever (2), used (2), direction (2), prelude (2), identify (2), carries (2), detail (2), probabilistic (2), clever (2), multiplying (2), cancelling (2), polynomials (2), single (2), warmup (2), respect (2), intuitively (2), expect (2), discussing (2), studied (2), studies (2), generalisation (2), famous (2), feature (2), tight (2), increment (2), stronger (2), obtained (2), appropriate (2), attempt (2), made (2), choose (2), karamata (2), forest (2), compound (2), str65 (2), bj20 (2), berard (2), juillet (2), 2020 (2), addressing (2), assumption (2), genuinely (2), requires (2), relax (2), support (2), touched (2), sizes (2), chord (2), interval (2), quantity (2), verify (2), corollary (2), proves (2), yes (2), heights (2), checking (2), thus (2), regardless (2), feels (2), forever (2), asking (2), outlined (2), project (2), concerning (2), determine (2), furthermore (2), upper (2), ended (2), research (2), forms (2), binary (2), exam (2), conditions (2), actual (2), challenge (2), simultaneously (2), replaced (2), starts (2), 999 (2), scenario (2), route (2), add (2), punchline (2), quick (2), complexity (2), congruent (2), perfectly (2), suffice (2), themselves (2), abc (2), focusing (2), bear (2), clean (2), convinced (2), conclude (2), attacking (2), coherently (2), masterclass (2), reflecting (2), session (2), matches (2), easier (2), turns (2), capture (2), inductive (2), theoretical (2), routes (2), credit (2), thirty (2), twelve (2), deductions (2), multiples (2), constructing (2), 100 (2), finding (2), recover (2), factorised (2), product (2), beyond (2), explicitly (2), wasn (2), involved (2), supposed (2), entirely (2), framing (2), valuable (2), attempting (2), digesting (2), reasons (2), returning (2), thursday (2), roughly (2), pell (2), antipode (2), careful (2), contestant (2), hoping (2), move (2), disjoint (2), replace (2), entries (2), were (2), together (2), expression (2), imply (2), generalise (2), amongst (2), monic (2), immediate (2), topics (2), necessarily (2), insight (2), derive (2), consequently (2), ends (2), force (2), goes (2), concurrency (2), although (2), sheet (2), welcome (2), subsequent (2), available (2), contact (2), mallows (2), randomly (2), shows (2), truncation (2), presented (2), independence (2), situations (2), intersections (2), implies (2), vanishing (2), intersection (2), makes (2), priori (2), closed (2), neat (2), lecture (2), currently (2), integrable (2), fundamentals (2), integration (2), exclude (2), footnotes (2), seem (2), limits (2), intervals (2), going (2), specific (2), back (2), approximate (2), footnote (2), arbitrarily (2), briefly (2), matter (2), attention (2), constructions (2), consequence (2), whole (2), ranges (2), finally (2), parts (2), needs (2), details (2), asked (2), plausible (2), simply (2), covered (2), minutes (2), global (2), distinction (2), shallow (2), obtuse (2), hexagon (2), marks (2), fault (2), affected (2), minimum (2), 250 (2), early (2), principles (2), preparing (2), broader (2), intermediate (2), cxy (2), assuming (2), remark (2), discover (2), minus (2), included (2), carefully (2), recursion (2), misreading (2), asymptotic (2), wish (2), residues (2), language (2), exercises (2), hand (2), factorisation (2), emphasise (2), bullet (2), eliminate (2), spoilers (2), answers (2), algebraic (2), aime (2), reversible (2), manipulations (2), family (2), teachers (2), secondary (2), olympiads (2), yeo (2), skip (2), name, loading, collapse, bar, report, manage, subscriptions, site, privacy, account, join, 220, subscribers, subscribed, subscribe, weak, team, preferential, attachment, perron, frobenius, optional, multiplicative, modular, chain, linyi, deviations, hitting, giant, exponential, tail, erdos, renyi, equilibrium, eigenvector, donsker, cramer, blumenthal, bijection, opens, tab, akismet, 626, 734, 224, hits, stats, jan, 2026, tom, lovering, tim, gowers, thesis, whisperer, aftermatter, terry, tao, renan, gross, gil, kalai, evan, chen, cp4space, correlated, blame, analyst, precision, andrew, gelman, blogroll, uncategorized, travel, 159, calculus, sle, maps, graphs, percolation, chains, interacting, particle, systems, gff, iii, revision, game, topology, spectral, categories, select, month, twin, presentation, centres, radii, thoughtful, flexible, distraction, harm, informal, stage, black, eligible, theorems, tangents, alternate, segment, characterise, literal, contender, lot, lifted, cropped, pointing, touches, formal, tell, hopefully, alone, viewing, maybe, shortcuts, narrow, pool, candidates, proposals, indicating, sanity, removed, breaking, chance, consists, pretty, remainder, division, fortunately, wider, candidate, major, considerable, warmly, encouraged, talk, oucolleague, vicky, neale, discusses, accessible, enjoyed, contain, saved, attempted, planning, yesterday, bmos, reproduced, replies, ohlin, 1969, dispersion, reinsurance, 1963, ben, hamou, peres, salez, bauerle, muller, 2005, consistency, 1967, dependent, letting, submartingale, achieved, orderings, exchanged, calculate, introduced, exotic, heuristic, precise, underpins, economic, relative, contribution, losses, thresholds, easily, having, according, personal, tried, reproduce, fail, randomness, meshes, coarsen, conditioning, depend, taken, notational, convenience, unambiguous, adapting, economics, actuarial, fields, criterion, danger, observer, aversion, inform, unsurprisingly, decomposition, casual, struggle, happened, sentence, treats, demonstrated, fkg, pre, reduce, cauchy, schwarz, said, little, rewrite, knowing, impediment, interpreted, determining, among, yet, preliminary, remarks, heavily, multi, moments, vanilla, settings, literature, addresses, potted, summary, proved, sections, merely, outlines, developed, choices, applying, relies, finiteness, estimate, boundedness, marginals, 1965, ecp, guaranteed, promising, avenue, established, completing, finish, people, visual, copy, canonical, adjust, pass, expectation, solves, brushes, generation, subclass, drawn, hypothesis, shall, warm, genesis, associated, interpretation, lost, scaling, happily, tied, corners, interested, her, preprint, subset, maximises, reaching, refined, critical, subcritical, typically, simplified, lying, dormant, revisited, posed, high, flying, doctoral, donderwinkel, revived, relevance, potpourri, expansion, impressed, rearrange, blocks, sppp, reducing, definiteness, string, introduces, afterwards, doomed, significance, hopeless, play, semi, intuitive, arrive, reasonably, minimal, relationship, reiterate, unnecessary, satisfactory, bisector, arc, midpoint, overblown, calculating, bisectors, centre, textbook, massively, cleans, firstly, ignore, blue, demonstrating, obscure, ones, illustrate, introducing, 120, regards, moral, ideally, keeps, track, determined, attempts, littered, demonstrate, notate, wherever, keep, explores, unclear, characterisations, roles, symmetry, got, offers, younger, gets, job, done, blogged, regularly, series, played, vice, versa, figuring, ish, fluent, manipulating, rational, decide, tiresome, backwards, reformulate, configurations, settles, divide, everything, 1009, 1010, complex, effectively, lend, multiply, facing, temptation, else, experiments, mistakes, dealing, strategy, justification, fine, fails, percentages, win, head, scratching, winning, nifty, rigorous, speculate, wins, eleven, rate, rates, things, aiming, 240, 288, 336, 360, 384, recipe, remember, seek, 2nm, strictly, algebraically, neatest, symmetrically, exhibits, eighteen, finished, impractical, uninteresting, needing, decompose, desired, missing, months, dominated, logistical, academic, refreshing, traditional, marking, stayed, 2000, pupils, overseas, friday, composition, cards, dimension, role, inducing, produce, seemed, task, partition, difficult, tasked, piece, odds, evens, extract, mentioning, bearing, divisibility, appears, varies, therefore, deduce, rearranging, due, szego, treated, heavier, concepts, knows, consult, davenport, lewis, schinzel, acta, arith, 1964, quadratics, establishes, degree, opportunity, reflect, image, contains, technique, previously, reduces, establishing, typical, myself, setters, origin, story, cleanest, reversal, collinear, quad, quads, looks, writer, insisting, fourth, chose, towards, quadruple, built, pop, waste, deduced, perpendiculars, sides, acpb, confirms, extending, hence, diameter, intersecting, denoted, explain, devised, diametrically, meets, lay, depicted, put, snakes, ladders, urn, multitype, harmonic, covariance, birthday, supervisors, anyone, instructors, please, receive, download, cambridge_extension_problems, supplement, inherited, james, norris, lecturers, prepared, enthusiastic, simplifications, graduate, gentle, colouring, emile, implausible, asymptotically, toy, working, regarding, trap, separated, changes, applies, pointwise, popular, models, properties, supervised, allows, alternatively, prevent, entry, reflection, limiting, occurs, iteratively, identity, inserting, position, realm, asserts, typing, keyboard, type, william, shakespeare, involve, decaying, speeds, overlap, classic, diverges, continuity, assumed, tractable, counts, occur, reflected, comparative, state, happy, admit, confused, prefer, stick, safety, flexibility, whichever, individual, literally, increases, richer, closer, title, axioms, demand, provides, meaningful, extremely, outcomes, thornier, uncountable, axiom, unions, holding, noticed, remit, allowance, king, builds, ground, week, matters, surprise, progression, pedagogical, schedule, benefits, experience, loosely, expressed, rounded, run, operation, commute, demanding, ideal, undefined, disappointing, reality, ending, versus, guarantee, approximation, concrete, help, unfortunately, notwithstanding, mentioned, committing, infimum, sup, attained, issue, suggestion, prompted, negativity, jointly, allow, relaxing, restriction, reals, analogue, revisit, happens, leaving, restrict, continuous, discontinuities, framework, truncated, integrability, tweak, definitions, alternatives, mostly, onto, contributions, implicitly, compact, provided, informally, bonus, approximations, paragraph, tempting, abstract, heavy, operator, unsurprising, recurring, theme, intuition, area, emphasising, contrast, splitting, recap, someone, deserves, address, directions, summarise, differences, usage, familiar, space, kcl, cover, higher, formality, introductory, courses, chase, spoil, wasted, effects, affects, nearby, average, absolutely, advantage, claim, justifying, correct, octagon, instructive, notice, regular, decorate, participants, scored, fair, astray, pursuing, articulate, unlikely, ran, rbbr, index, properly, speaking, anticlockwise, justify, listing, indexes, groups, repeating, colours, guess, live, commit, haven, missed, totally, unrelated, explaining, yellow, white, houses, mog, unusual, conjectured, smallest, template, spending, hours, competitions, devoting, hour, come, submitted, separate, intriguing, diagonals, kites, accidentally, bcxy, reflections, big, hole, axbz, explains, match, efficiently, compasses, observe, combined, abx, abxz, avoiding, mindless, solved, identifying, possibly, occasional, geometrical, near, top, practising, subject, reference, bring, investigation, noticing

Text of the page (random words):
m 6 this year more participants than usual scored close to full marks on bmo1 but i don t think it is fair to say that this particular problem was too easy the harder part of the problem proving that it is necessarily true for n odd is relatively short that doesn t mean that it s obvious and there are many ways to go astray i personally spent 20 minutes on the n odd case pursuing ideas that are hard to articulate but certainly did not work it just means that it s relatively unlikely that a contestant had the main idea but ran out of time to write it it s crucial to establish the correct answer by trying various constructions it s possible that the octagon n 8 was more instructive than the hexagon n 6 it s crucial to notice that if you start with a regular k gon for any k 2 you can decorate each side with congruent shallow isosceles triangles to get a 2k gon that satisfies the length condition but doesn t have equal angles one advantage of taking the isosceles triangle to be very shallow is that we can claim that when is small enough the 2k gon is convex and that the obtuse angle in the triangle is greater than the total obtuse angle at the vertices of the k gon without formally justifying this with algebra of course this approach does not make sense for constructing an n gon when n is odd but this is absolutely not a proof that there is no such construction for n odd however in the context of an olympiad problem at this level it is reasonable evidence that the statement might be true for n odd the condition given about the lengths is a local condition meaning that each condition only affects nearby vertices when thinking about proving this result one needs to consider whether we will find a local conclusion eg two consecutive angles are equal or a global conclusion eg every angle is less than or equal to the average angle this distinction is less clear when n is small of course i won t spoil the end of the problem except to say that i wasted my 20 minutes thinking about global effects whereas actually the main route involves a neat observation of the local situation posted in british maths olympiad euclidean geometry olympiad combinatorics olympiad mathematics tagged angle chase bmo1 bmo1 2023 british mathematical olympiad combinatorics euclidean geometry olympiad permutation recurrence relation 1 reply why do we need the lebesgue integral posted on november 16 2023 by dominicyeo reply i m currently lecturing the course fundamentals of probability at kcl where we cover some of the measure theory required to set up probability with a higher level of formality than students have seen in their introductory courses by this point we have covered sigma algebras measures and probability spaces including the fact that not all subsets of are borel measurable measurable functions which in probability spaces are random variables more recently we ve covered the construction of the lebesgue integral for measurable functions for some general measure space in this post i ll summarise briefly this construction and some key differences in context and usage between this and the more familiar riemann integral but i ll also aim to answer the question why do we need to set up the lebesgue integral like this why can t we do it directly or more simply someone asked essentially this question after the lecture and i think it deserves an answer to address this i ll look a handful of plausible alternative directions for defining integration and seeing where they go wrong construction of the lebesgue integral but first we have to recap what the construction is it goes in three steps construction for simple functions which only take finitely many values and all these values are non negative construction for measurable non negative functions by considering a monotone limit of simple functions construction for general measurable functions by splitting into positive and negative parts it s worth emphasising that for all of these steps the integral is defined directly over the whole set e this is an immediate contrast with the improper riemann integral for functions where the integral has to be defined as a limit of integrals over finite ranges adding a few details a simple function has the form where are non negative coefficients and are measurable sets note that the sum is finite then the integral of is defined to be this matches our intuition if we are thinking of functions with an idea of integrals as area under a graph but it is just a definition question we will see shortly why it s only relevant to consider non negative coefficients but initially we might ask why it needs to be a finite sum in general when sums are finite then they don t need to be defined as a limit furthermore they definitely exist if we allowed infinite sums in the definition of simple functions we would then need to exclude pathologies like which is infinite everywhere 1 this will be a recurring theme of this post an optimal definition only takes a limit when it s certain that the limit exists anyway the arguments are sometimes a bit notation heavy but in the setting of simple functions we can prove directly that the integral is a linear operator and so satisfies and various other unsurprising but important results following this we define the integral for general measurable functions taking only non negative values via a limit of simple functions specifically while it would be tempting to define as the limit of the integrals of a specific sequence of simple functions approximating for example defining by rounding down to the nearest multiple of the more abstract definition turns out to be more useful in proofs 2 using definition we derive the monotone convergence theorem informally this says that monotone limits of non negative measurable functions respect integrals more formally when a e then as a bonus we immediately recover that the limit of the integrals of the rounding down approximations in the previous paragraph is the limit of the original function finally we extend to general measurable functions potentially taking both positive and negative values we identify the positive and negative parts of and define the integral provided that at least one of the rhs integrals is finite the point is that it s well defined to write or but it s not well defined to study it might seem annoying that a consequence of this is that functions such as are then not integrable over even though they have a finite improper riemann integral but this is a direct consequence of defining the lebesgue integral directly onto the whole set e any thoughts about positive and negative contributions cancelling out are implicitly or explicitly thinking about as a limit of compact ranges 3 ideas for alternative constructions given the non integrability of certain functions as above one might consider whether it s possible to tweak the definitions to avoid this here is a list of reasonable alternatives to the standard order of construction described above in each case i ll draw attention to something that might go wrong focusing on the case but the issues mostly generalise truncation of range we ve touched on this earlier but let s briefly revisit what happens if we define integrals as a limit of intervals over finite range leaving aside the question of whether this is possible for more general spaces see 3 we already have issues with this approach over note that once we restrict to finite range and continuous functions or finitely many discontinuities it doesn t matter if we are using riemann or lebesgue framework but when studying a function like we have truncated results like and so taking the limit would define the integral of f over to be zero while taking the limit would give a different answer so this certainly wouldn t work as a general definition in practice for improper integrals one generally studies with the same restriction on taking as we discussed earlier in this post however this split into the two half lines is very much a feature of the reals even just in there is no easy analogue to this split approximating all functions from below what about if we try to approximate all measurable functions from below by simple functions after relaxing the condition that the coefficients of a simple function have to be non negative well if a function takes arbitrarily negative values eg all values in then it just can t be bounded below by a function which only takes finitely many values and if we allow simple functions to have negative coefficients and support on infinitely many sets this is even worse than the situation discussed in footnote 1 already at this point we would already have simple functions for which the integral isn t defined such as taking for and for it s not going to work very well to study the supremum of a set some of whose values are not defined jointly approximating from below above this is the suggestion that prompted me to write this post suppose we relax the non negativity but not the finite sum condition for a simple function then approximate by simple functions so that and a e that is bounds f from below when f is positive and from above when f is negative the issue here is that it s not clear how to turn this into a definition we can t write because the sup would be attained by considering simple functions g which are zero whenever f is negative and it s not well defined to replace the supremum with a limit it s not a sequence after all if you try and split it as a supremum over the range where and an infimum over the range where then you are actually back to the original definition of the general lebesgue integral and going for a very concrete sequence doesn t help either unfortunately if we define by rounding down up depending on whether is positive negative to the nearest multiple of we could define notwithstanding some of the issues we mentioned earlier about committing to an specific approximating sequence however since the are not monotone in n there is no guarantee that this limit exists and indeed we can construct examples of where the limit does not exist consider for example the intervals and the function so then the approximation and thus which is 1 when n is odd and 0 when n is even with the true definition of the lebesgue integral is undefined which might seem equally disappointing but in reality is more clear than ending up in a situation where you get different limits depending on whether you round down to the nearest multiple of versus rounding down to the nearest multiple of footnotes 1 it s not impossible to exclude such pathologies for example by demanding that the are disjoint but adding constraints to the definition of simple function is far from ideal if we need to work with them for proofs for example under it becomes less obvious that the sum of two simple functions is simple 2 if we define to be rounded down to the nearest multiple of and similarly for then we immediately run into problems when trying to prove that as the rounding down operation doesn t commute with addition 3 and there are measure spaces which are not sigma finite for which e cannot be expressed as a countable union of finite measure sets but we still want a theory of integration over such spaces posted in analysis measure theory university teaching tagged borel measure fundamentals of probability integrable function lebesgue integral measurable function measure theory monotone convergence theorem monotone limit sigma finite simple function leave a reply borel cantelli lemmas posted on october 6 2022 by dominicyeo reply i m currently lecturing my first course at king s which builds measure theory from the ground up to the construction of the lebesgue integral along with some more probabilistic topics in this second week we have been discussing various matters related to the construction of sigma algebras it took me slightly by surprise that this ends up being the natural moment during an undergraduate s progression through probability to introduce the borel cantelli lemmas it s not hard to imagine a pedagogical schedule in which they appear earlier but equally one benefits from seeing them with a bit more mathematical experience even if they are only very loosely related to the construction of the lebesgue integral anyway i noticed that i have not written about these results on the blog before there are a couple of neat applications that lie slightly beyond the remit and time allowance of the lecture course so they are presented at the end context events holding eventually and infinitely often recall that in the notation of probability spaces a sigma algebra provides a set of events for which it is meaningful and consistent to define a probability measure this is extremely uncontroversial in the setting where the set of outcomes is finite but thornier in the setting where is uncountable for example 0 1 or the key axiom for sigma algebras involves countable unions if is a sequence of events in we must also have since the axioms also demand that is closed under taking complements it only takes a few steps to show that countable intersections are also valid ie also it is uncontroversial to note that the main initial motivation for considering the events are that they lie between the intersection and the union in terms of containment infinitely often is relatively uncontroversial just meaning for infinitely many values of n eventually is less clear on an initial reading perhaps eventually always would be better and closer to the title of this blog the meaning is always except for finitely many n which implies that there is some threshold n such that it holds for all n n in set notation infinitely often or i o means that must hold for every n and so the event similarly eventually means that there must be an n for which holds so the event we end up with the richer containment sequence these individual containment relations may not be obvious a priori in this form probably the most clear is the central containment that if an holds for all except finitely many n then it holds for infinitely many n the right most says that if it holds infinitely often then it holds at least once for the left most containment note that is literally the same as the n 1 case of so taking a union over other values of n only increases the set it s also worth thinking about the complements note that and versions of these apply to eventually i o also in particular and so in any calculation or application there is flexibility to study is this is more convenient as we will see it makes sense to focus on whichever of has vanishing probabilities as comment it is standard to write and i am happy to admit i get these confused with non vanishing probability and prefer to stick to eventually i o or the full union intersection statements for safety borel cantelli lemmas the borel cantelli lemmas establish some conditions under which the event almost surely holds or almost surely doesn t hold we ll state them now ...