# Questions tagged [measure-theory]

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

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### Weak Convergence for a continuous function without compact support

Let $X$ be an open subset of $\mathbb R^n$ and let $\Omega$ be a relatively compact, open subset of $X$. Let $\{ \mu_n\}$ be a sequence of positive measures that converge weakly to the positive ...
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### Complex valued Stieltjes integrals : If $f\in\mathcal{R}(\alpha)$, do we have $\mathfrak{Re}(f), \mathfrak{Im}(f)\in\mathcal{R}(\alpha)$?

Given a bounded complex function $\alpha:[a,b]\to\mathbb{C}$, we can define the Riemann-Stieltjes integral of $f:[a,b]\to\mathbb{C}$ (also bounded) in a way that is very much analogous to the usual ...
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### Show that if $\{x: f(x)\neq g(x)\}, \{x: g(x)\neq h(x)\}\in\mathfrak{M}$ then $\{x: f(x)\neq h(x)\}\in\mathfrak{M}$.

The Problem: Show that if $\{x: f(x)\neq g(x)\}, \{x: g(x)\neq h(x)\}\in\mathfrak{M}$, then $\{x: f(x)\neq h(x)\}\in\mathfrak{M}$, where $f, g, h$ are measurable functions. The problem arises from ...
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### approximated continuity

what is an example of measurable function $f$ :$\mathbb R^n$ $\to$ $\mathbb R$ with a point $x_0$ that is not a point of approximated continuity?
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### For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X$, and a function $g$, is $g(X)$ automatically measurable?

For the probability triple $(\Omega, \mathcal{F}, \Bbb{P})$, a random variable $X: \Omega \to D$, and an arbitrary function $g: D \to E$, is $g \circ X$ also measurable and thus a random variable? I ...
1 vote
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### Approximating $\mathcal X \subset \mathbb R^n$ with the union of disjoint hyperrectangles

Let $\mathcal X$ be a bounded subset of $R^n$ that is generated by a set of linear inequalities. For example, let ${\bf x} = (x_1, x_2, x_3, x_4, x_5)^\intercal \in \mathcal X$ iff \begin{align*} 0 &...
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### If $\mu_i \to \mu$ weakly, then do we have $\operatorname{supp}(\mu_i) \to \operatorname{supp}(\mu)$ in Hausdorff distance?

Suppose $\mu_i$ are Radon measures with support in $\mathbb{R}^{n}$. Is it true that if $\mu_i \to \mu$ weakly then their supports must converge in Hausdorff distance?
1 vote
### Inverse image of and $L^p$ function has null measure
Let $f \in L^{p}(\mathbb{R}^n)$ (maybe just mensarable). Is it true that $f^{-1}(\{c\})$, has zero Lebesgue measure, where $c$ is a constante. I know in the case $f$ is smooth this is true, because \$...