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There is nothing mathematically wrong with your notation. However, I don't like it, because $Z'$ suggests that you are taking a derivative with respect to the background randomness. I would rather w …

Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions. Is there a direct ch …

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in …

A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spaces, but to the best …

In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$. Will Sawin described the mon …

Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space? Here is one natural …

Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\sig …

Thomas Bloom is right: the proof of the usual Chebyshev inequality can be easily adapted to the higher moment case. Rather than looking at the statement of the theorem and being satisfied with it, h …

This is a sequel to another question I have asked. The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; someti …

In addition to the excellent answers above, I also suggest the nice survey Oxtoby, Ergodic Sets (Zbl 0046.11504, MR47262, DOI: 10.1090/S0002-9904-1952-09580-X). Introduction. Ergodic sets were introd …

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far fr …

The following question came up in a discussion with my advisor: Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-alg …

Arkadiusz gives the answer in the case of two independent Gaussians. A simple technique to reduce the correlated case to the uncorrelated is to diagonalize the system. The intuition which I use is t …

Let $X_i$ be a sequence of identically-distributed random variables with finite-range dependence (i.e. there exists $I$ such that if $|i-i'| \ge I$, then $X_i$ and $X_{i'}$ are independent), and a fin …

In Appendix B of his Uniform Central Limit Theorems (1999), Dudley writes: It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable cardinals, …