· 5 min
The exact condition for convergence in the mean. Uniform integrability, its characterisation by uniform absolute continuity, and the Vitali theorem that convergence in measure upgrades to L-one convergence exactly when the sequence is uniformly integrable.
· 5 min
The Banach spaces of integrable powers. Young's inequality, the Holder and Minkowski inequalities that make the p-norm a norm, and the completeness theorem that promotes every L-p to a Banach space, the family of which L-squared is the one Hilbert member.
· 5 min
Filtrations, stopping times, and the predictable sigma-algebra that encodes non-anticipation. We prove the basic properties of the stopping-time sigma-algebra and identify predictable processes with the measurable closure of the simple integrands.
· 5 min
The modes of convergence for random variables and the two theorems that govern sample averages. We prove the Markov and Chebyshev inequalities, Borel-Cantelli, a strong law under a fourth moment, and the central limit theorem by characteristic functions.
· 5 min
Conditional expectation defined by its averaging property, shown to exist via Radon-Nikodym, and identified with orthogonal projection in L^2. We prove the tower property, the pull-out rule, and conditional Jensen.
· 8 min
The factorisation that makes randomness combine. Independence of events, sigma-algebras, and random variables, the equivalence with a product law, the factorisation of expectations through Fubini, the Borel-Cantelli lemmas, and the Kolmogorov zero-one law for tail events.
· 4 min
Absolute continuity, equivalence, and the density that connects two measures.
· 7 min
The Lebesgue integral built from simple functions, and the three convergence theorems that make it usable. We prove monotone convergence, deduce Fatou and dominated convergence, and record the L^p inequalities.
· 6 min
Probability as measure theory with total mass one. The probability space, random variables and their laws as pushforward measures, expectation as the integral, the change of variables that computes it from the law, and the Markov, Chebyshev, and Jensen inequalities.
· 7 min
When a double integral equals an iterated one. The product sigma-algebra and the measurability of sections, the construction of the product measure, and the Tonelli and Fubini theorems that exchange the order of integration for nonnegative and for integrable functions.
· 5 min
The two model infinite-dimensional Hilbert spaces, the square-integrable functions and the square-summable sequences, and the Riesz-Fischer theorem that makes them complete.
· 8 min
How size is assigned to sets. Sigma-algebras and measures, continuity along monotone limits, closure of measurable functions under pointwise limits, the Caratheodory extension theorem that turns an outer measure into a measure, and the construction of Lebesgue measure.