Algebraic probability spaces

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As laid out in the foundational work of Kolmogorov, a classical probability space (or probability space for short) is a triplet $latex {(X, {mathcal X}, mu)}&fg=000000$, where $latex {X}&fg=000000$ is a set, $latex {{mathcal X}}&fg=000000$ is a $latex {sigma}&fg=000000$-algebra of subsets of $latex {X}&fg=000000$, and $latex {mu: {mathcal X} rightarrow [0,1]}&fg=000000$ is a countably additive probability measure on $latex {{mathcal X}}&fg=000000$. Given such a space, one can form a number of interesting function spaces, including

  • the (real) Hilbert space $latex {L^2(X, {mathcal X}, mu)}&fg=000000$ of square-integrable functions $latex {f: X rightarrow {bf R}}&fg=000000$, modulo $latex {mu}&fg=000000$-almost everywhere equivalence, and with the positive definite inner product $latex {langle f, grangle_{L^2(X, {mathcal X}, mu)} := int_X f g dmu}&fg=000000$; and
  • the unital commutative Banach algebra $latex {L^infty(X, {mathcal X}, mu)}&fg=000000$ of essentially bounded functions $latex {f: X rightarrow {bf R}}&fg=000000$, modulo $latex {mu}&fg=000000$-almost everywhere equivalence, with $latex {|f|_{L^infty(X, {mathcal X}, mu)}}&fg=000000$ defined…

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