# Tagged Questions

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

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### Continuous functions are locally integrable?

If $K\subset\mathbb{R}$ is compact and $f:K\rightarrow\mathbb{R}$ continuous then $f\in\mathbb{L}(K)$. In other words $f$ is integrable in $K$. So far i know that since $f$ is continuous then $f(K)$ ...
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### What is a Dynkin system? ($\lambda$-system)

Until recently, all my knowledge of measure theory and Lebesgue integration are from Rudin's book, which focuses solely on the Lebesgue measure, its construction and nothing else. I have just put my ...
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### Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
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### Application of Dominated Convergence Theorem help finding a Dominating function

$$\lim_{n\to\infty}\int_0^\infty \frac{n\sin(x/n)}{x(1+x^2)}$$ I wish to use the Lebesgue Dominated Convergence theorem to solve this, but I'm having trouble finding a dominating function, $g(x)$. ...
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### Is it a change of variable?

Hi everyone: In a book I am reading, they make a sort of "substitution" like this: let $B(0,R)$ be a ball in $\mathbb{R}^{N}$ $(N\geq2)$ and $f$ a locally integrable function. Let $\mu$ be a finite ...
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63 views

### Measure on $\omega$ defined in the generic extension by an atomless measure algebra is atomless

Work in Cantor space with standard probability measure $m$. Suppose we are given a sequence of measurable sets $\bar{A}=\langle A_n : n\in \omega\rangle$ and a non-principal ultrafilter $U$ and the ...
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### show that continuous functions on $\mathbb{R}$ are measurable

I am trying to show this using the theorem: A function $f: \Omega \to \mathbb{R}$ is measurable if and only if $f^{-1}(E) \in \mathcal{F}$ for all borel sets $E$. The proof to show a continuous ...
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### f left continuous & strictly increasing; B Borel $\implies$ f(B) Borel (or at least Lebesgue Measurable)?

How's it going? In an attempt to use the Radon-Nykodym theorem to bulldoze through the admission of measures by bounded variation & monotonic functions (sidestepping all that Caratheodory ...
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### Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

I am reposting a question from Math Overflow, because it seems it gets no attention. Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak ...
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### In Egorov's Theorem, is almost everywhere same as point-wise?

I am studying about Egorov's Theorem. My teaching assistant said to me that Egorov's theorem is roughly like the following statement: Under two conditions which are $|E|\lt+\infty$ and ...
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### If $\{T_n\} \to T$ and $\{u_n\} \to u$, then $\{T_n(u_n)\} \to T(u)$.

Let $X$ and $Y$ be normed linear spaces. Define $$L(X,Y) = \{T:X \to Y \ \big | \ T \text{ is bounded}\}.$$ Let $\{T_n\} \to T$ in $L(X,Y)$ and $\{u_n\} \to u$ in $X$, then $\{T_n(u_n)\} \to T(u)$ ...
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