Real Analysis and Probability

algebra axiom B₁ Borel measurable Borel o-algebra Borel set bounded Brownian motion called Cartesian product Cauchy characteristic function closed set compact sets complete continuous function Corollary countably additive defined definition dense disjoint distribution function equivalent ergodic example extended finite measure finite set fn(x follows function f given Hausdorff space Hilbert space Hint implies independent inequality infinite inner product integral intersection Lebesgue measure Lemma Let f linear Markov martingale Math measurable function measurable sets measure space metric space non-empty nonnegative norm open sets orthonormal probability measure probability space Problem product topology proof of Theorem Proposition proved random variables real numbers real-valued function separable metric space sequence Show smallest o-algebra submartingale subsets Suppose Theorem Let topological space uncountable uniformly union unique values vector space x₁