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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254A, Notes 0: A review of probability theory

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In preparation for my upcoming course on random matrices, I am briefly reviewing some relevant foundational aspects of probability theory, as well as setting up basic probabilistic notation that we will be using in later posts. This is quite basic material for a graduate course, and somewhat pedantic in nature, but given how heavily we will be relying on probability theory in this course, it seemed appropriate to take some time to go through these issues carefully.

We will certainly not attempt to cover all aspects of probability theory in this review. Aside from the utter foundations, we will be focusing primarily on those probabilistic concepts and operations that are useful for bounding the distribution of random variables, and on ensuring convergence of such variables as one sends a parameter $latex {n}&fg=000000$ off to infinity.

We will assume familiarity with the foundations of measure theory; see for instance these…

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