Courses / Mathematics / Measure, Lebesgue integration and the probability bridge Mathematics · unit version 1 Measure, Lebesgue integration and the probability bridge A proof-based advanced undergraduate sequence constructing measures and integrals, proving convergence and Lp theorems, and connecting density and conditional-expectation arguments to probability.
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Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
How do pointwise and uniform convergence differ? Integral limit theorems need hypotheses stronger or different from pointwise convergence alone.
Compare reasoning and find a refresher Pointwise convergence permits the index cutoff to depend on the point. Uniform convergence requires one cutoff for every point in the domain. A shrinking spike can converge pointwise without uniform convergence.
Review convergence quantifiers
How do you prove equality between two sets? Sigma-algebra and measurability proofs repeatedly require both inclusions.
Compare reasoning and find a refresher Show every member of the first set belongs to the second and conversely. For countable operations, state the quantified membership conditions rather than checking only a few examples.
Review set inclusion proofs
Why can a finite-measure assumption matter when taking decreasing intersections? Continuity of measures is proved using complements and subtraction of finite quantities.
Compare reasoning and find a refresher A decreasing sequence can have empty intersection while every set has infinite measure, for example tails [n,∞) of the real line. One finite-measure starting set avoids subtracting infinity from infinity.
Review completeness and bounds
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Measure and integration: construction, convergence and probability
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Lesson details and objectives Construct generated sigma-algebras from observations.
Construct generated sigma-algebras from observations. Distinguish countable closure from arbitrary unions. Identify atoms in a finite measurable space. 6 practice tasks.
Lesson details and objectives Prove continuity from below and qualified continuity from above.
Prove continuity from below and qualified continuity from above. Use disjointification to derive subadditivity. Diagnose the finite-measure hypothesis. 6 practice tasks.
Lesson details and objectives Prove an interval-cover outer measure is subadditive.
Prove an interval-cover outer measure is subadditive. Compute interval and countable-set outer measures. Distinguish an outer measure from a measure. 6 practice tasks.
Lesson details and objectives Prove closure of Carathéodory measurable sets.
Prove closure of Carathéodory measurable sets. Derive countable additivity on that sigma-algebra. Explain completeness of the resulting measure. 6 practice tasks.
Lesson details and objectives Construct an outer measure from a premeasure.
Construct an outer measure from a premeasure. Prove agreement on the original algebra. Use a pi-lambda argument for sigma-finite uniqueness. 6 practice tasks.
Lesson details and objectives Construct Lebesgue measure from interval length.
Construct Lebesgue measure from interval length. Prove outer and inner regularity in finite regions. Use regularity to control approximation errors. 6 practice tasks.
Lesson details and objectives Describe the completion of a measure.
Describe the completion of a measure. Prove a Vitali-type nonmeasurability contradiction. Explain the role of countable additivity and choice. 6 practice tasks.
Lesson details and objectives Prove threshold criteria for measurability.
Prove threshold criteria for measurability. Construct increasing dyadic simple approximants. Preserve measurability under limits and arithmetic. 6 practice tasks.
Lesson details and objectives Prove representation independence for simple integrals.
Prove representation independence for simple integrals. Define the nonnegative integral by a supremum. Derive monotonicity and null-set invariance. 6 practice tasks.
Lesson details and objectives Prove the monotone convergence theorem from simple functions.
Prove the monotone convergence theorem from simple functions. Apply it to nonnegative series. Explain why nonnegativity and monotonicity matter. 6 practice tasks.
Lesson details and objectives Derive Fatou’s lemma using tail infima.
Derive Fatou’s lemma using tail infima. Apply lower semicontinuity to integral estimates. Construct a strict-inequality example. 6 practice tasks.
Lesson details and objectives Define signed integrals without infinity minus infinity.
Define signed integrals without infinity minus infinity. Prove dominated convergence from Fatou. Distinguish pointwise convergence from integral convergence. 6 practice tasks.
Lesson details and objectives Prove Egoroff’s theorem on a finite measure space.
Prove Egoroff’s theorem on a finite measure space. Derive Lusin approximation on a bounded interval. Explain why the exceptional set cannot always be removed. 6 practice tasks.
Lesson details and objectives Verify uniform integrability using tail bounds.
Verify uniform integrability using tail bounds. Prove the finite-measure Vitali convergence criterion. Diagnose bounded L1 families that fail it. 6 practice tasks.
Lesson details and objectives Prove measurability of sections by a monotone-class argument.
Prove measurability of sections by a monotone-class argument. Construct the sigma-finite product measure. State uniqueness and completion distinctions. 6 practice tasks.
Lesson details and objectives Prove Tonelli by indicators and monotone approximation.
Prove Tonelli by indicators and monotone approximation. Use it for infinite sums and tail integrals. Allow infinity without subtracting it. 6 practice tasks.
Lesson details and objectives Derive Fubini by positive and negative parts.
Derive Fubini by positive and negative parts. Compute an order-dependent counterexample. Interpret almost-everywhere section statements. 6 practice tasks.
Lesson details and objectives Prove the pushforward integration formula.
Prove the pushforward integration formula. Derive one-dimensional density transformations. Separate measure transport from a general Jacobian theorem. 6 practice tasks.
Lesson details and objectives Define Lp classes and essential bounds.
Define Lp classes and essential bounds. Prove Hölder from Young’s inequality. State equality and endpoint cases. 6 practice tasks.
Lesson details and objectives Derive Minkowski using Hölder.
Derive Minkowski using Hölder. Compare Lp norms on finite measure spaces. Explain why p below one gives a different geometry. 6 practice tasks.
Lesson details and objectives Prove completeness with a summable-difference subsequence.
Prove completeness with a summable-difference subsequence. Approximate Lp functions by simple functions. Use Lebesgue regularity for continuous approximation. 6 practice tasks.
Lesson details and objectives Prove Hahn decomposition for a finite signed measure.
Prove Hahn decomposition for a finite signed measure. Construct mutually singular Jordan parts. Compute total variation from those parts. 6 practice tasks.
Lesson details and objectives Prove the finite Radon–Nikodym theorem.
Prove the finite Radon–Nikodym theorem. Extend the result by sigma-finite localization. Establish uniqueness up to null sets. 6 practice tasks.
Lesson details and objectives Construct absolutely continuous and singular parts.
Construct absolutely continuous and singular parts. Prove uniqueness of the decomposition. Distinguish singular continuous measures from atoms. 6 practice tasks.
Lesson details and objectives Prove a one-dimensional weak maximal estimate.
Prove a one-dimensional weak maximal estimate. Derive Lebesgue differentiation by approximation. Identify Lebesgue points and null exceptional sets. 6 practice tasks.
Lesson details and objectives Prove indefinite integrals are absolutely continuous.
Prove indefinite integrals are absolutely continuous. Derive the fundamental theorem through measure densities. Separate bounded variation from absolute continuity. 6 practice tasks.
Lesson details and objectives Construct probability laws as pushforwards.
Construct probability laws as pushforwards. Extend independence from generating pi-systems. Relate independent laws to product measures. 6 practice tasks.
Lesson details and objectives Prove both Borel–Cantelli lemmas under stated hypotheses.
Prove both Borel–Cantelli lemmas under stated hypotheses. Extract almost-sure subsequences from convergence in probability. Construct a sequence separating convergence modes. 6 practice tasks.
Lesson details and objectives Construct conditional expectation using Radon–Nikodym.
Construct conditional expectation using Radon–Nikodym. Prove uniqueness and the tower property. Compute finite-partition conditional means. 6 practice tasks.
Lesson details and objectives Prove conditional Jensen with supporting affine functions.
Prove conditional Jensen with supporting affine functions. Derive Lp contraction. Prove the L2 orthogonal projection identity. 6 practice tasks.
Lesson details and objectives Derive the weak law for finite-variance independent samples.
Derive the weak law for finite-variance independent samples. Prove a maximal inequality for partial sums. Use dyadic blocking for the finite-variance strong law. 8 practice tasks.
Lesson details and objectives Use bounded continuous test functions to define weak convergence.
Use bounded continuous test functions to define weak convergence. Derive characteristic-function continuity and the CLT expansion. Prove the continuity theorem through tightness, extraction and uniqueness. 8 practice tasks.
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