Questions tagged [measure-theory]

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

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6
votes
1answer
95 views
+50

Must any $\phi \in \operatorname{Hom}_G(V, L^2(G))$ have continuous values?

Let $G$ be a compact group and $V$ a finite-dimensional vector space with a continuous $G$-action. Consider a linear map $\phi: V \to L^2(G)$ satisfying that for any $v \in V, h \in G$: $$ \phi(v)(...
4
votes
1answer
163 views
+50

Doubts on application of continuity definition and Dominated Convergence theorem

I quote Øksendal (2003). Let $\mathcal{V}=\mathcal{V}(S,T)$ be the class of functions $f(t,\omega):[0,\infty)\times\Omega\to\mathbb{R}$ such that $(t,\omega)\to f(t,\omega)$ is $\mathcal{B}\times\...
1
vote
1answer
93 views
+50

Doubts on procedure of approximation of bounded functions by means of bounded, continuous functions

I quote Øksendal (2003). My doubts along the below proof will be written in $\color{red}{\text{red}}$. Let $\mathcal{V}=\mathcal{V}(S,T)$ be the class of functions $f(t,\omega):[0,\infty)\times\...
0
votes
0answers
68 views
+50

Equivalent definition of inner Jordan measure (and why doesn't it work for Lebesgue inner measure?)

Let $M_k$ be an elementary set, $|M_k|$ - it's volume. $K$ is a cube of volume 1. Let $E \subset K$. Then inner Jordan measure $\mu^J_*(E)$ of set $E$ is defined as $\mu^J_*(E) = 1 - \mu^*_J(\...
2
votes
0answers
57 views
+100

A general but accessible version of the divergence theorem

The purpose of this question is to gather one/multiple statements of the divergence theorem that can cover most of the cases that one might encounter, say, in a standard PDE course (see the bottom of ...
1
vote
0answers
24 views
+50

Decomposition of the variation of a signed measure as $|\mu|(A) = \int_A |\frac{d\mu_{1a}}{d\mu_2}-1|d\mu_2 + \mu_{1s}(A)$, where $\mu=\mu_1-\mu_2$

Let $\mu_1$ and $\mu_2$ be two finite measures on $(\Omega, \mathcal{F})$. Let $\mu_1 = \mu_{1a}+\mu_{1s}$ be the Lebesgue decomposition of $\mu_1$ w.r.t. $\mu_2$, that is, $\mu_{1a} \ll \mu_2$ and $\...