Measure TheoryThis is different from other books on measure theory in that it accepts probability theory as an essential part of measure theory. This means that many examples are taken from probability; that probabilistic concepts such as independence, Markov processes, and conditional expectations are integrated into the text rather than relegated to an appendi. |
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absolutely continuous According to Theorem application Borel measurable bounded variation Co(R Co(S compact metric space compact separable metric conditional expectations context continuous function Coo(S coordinate functions corresponding countable additivity countable union defined definition dense disjunct union equivalence classes everywhere example finite measure space finite unions finite valued finitely additive following theorem Fourier function f Hilbert space implies increasing sequence indicator functions inequality infinite sequence intersections Lebesgue measure Let f lim inf lim sup locally compact martingale measurable from S,S measurable function measurable sets monotone function monotone increasing function mutually independent null set orthogonal outer measure probability measure probability space Proof pseudometric space random variables right continuous right semiclosed intervals S₁ satisfying Section semicontinuous separable metric space sequence f sequence of measurable set function signed measure strictly positive subadditive submartingale supermartingale Suppose supremum trivial uniform integrability uniformly vanishes vectors