# Questions tagged [measure-theory]

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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### Calculating PDF of a function of random variables via dirac delta integral

I came across some papers 1, 2 and others which use the following formula $p(y) = \int_{\mathcal{X}} \, d^{n}x \, \delta(F(\vec{x}) - y) \, p(\vec{x})$ where $p(y)$ is the PDF of the variable ...
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### Different measurability of Hilbert-space valued random variable

My question is motivated by this link. Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable. Now let $H$ be a ...
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### Pettis integral on locally convex space and seminorms

Let $E$ be a locally convex Hausdorff space, and $X$ be a locally compact Hausdorff space which we fix a positive Radon measure $\mu$. Assume that $f: X \to E$ is a function such that the Pettis-...
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### To show that characterization of a point mass distribution for a given function is a measure [closed]

Let $X$ be a nonempty set, and let $A$ be a $\sigma$-algebra on $X$. Let $x$ be a member of $X$. Define a function $\delta x: A \to [0,\infty]$ by letting $\delta x(A)$ be $1$ if $x \in A$ and letting ...
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### Is there more than one notion of algebras? [duplicate]

I know that an algebra is an algebraic structure, that can be seen as a vector space with a multiplication operation or as a ring with a vector space structure. However, in measure theory we define an ...
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### Hausdorff measure of uncountable dense subsets

As far as I know, the Hausdorff measure of a countable subset is zero (please correct me otherwise). Is it possible to say the same about the uncountable dense subsets? Is there a general statement ...
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