# Questions tagged [measure-theory]

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

8,116 questions with no upvoted or accepted answers
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### Time derivative of a measure and convolution

Suppose $\mu_t : [0, + \infty) \to \mathcal{P}(\mathbb{R}^n)$ is a curve of probability measures on $\mathbb{R}^n$. For each $\nu \in \mathcal{P}(\mathbb{R}^n)$ I define $\nu \ast \rho$ as the ...
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### Topology and $\sigma$-algebra on the space of probability measures

Let $(X,\mathbf{A})$ be a $\sigma$-algebra of sets, $\mathscr{M}(\mathbf{A})$ be the set of all $\mathbf{A}$-measurable functions $f\colon X\to [0,1]$, and $\mathscr{P}(\mathbf{A})$ be the set of all ...
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### limsup of continuous function is measurable.

I want to show that if $F$ is continuous on $[a,b]$ then $$\limsup_{h \rightarrow0, h>0} \frac{F(x+h)-F(x)}{h}$$ is measurable. By the definition of $\limsup$ we can write \begin{align} &\...
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### Is there a continuous and injective function $f: [0,1] \to \mathbb R^2$ such that the image has positive two dimensional Lebesgue measure?

Is there a continuous and injective function $f: [0,1] \to \mathbb R^2$ such that the image has positive two dimensional Lebesgue measure? I am not sure how to approach here. I think ine way to ...
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### Continuous function $f:\mathbb R\to \mathbb R$ such that $f(B)$ is not measurable for a Borel set $B$.

1) I saw in a book that there are $\mathcal C^1(\mathbb R)$ function $f:\mathbb R\to \mathbb R$ such that $f(B)$ is not measurable for $B$ a Borel set. I don't really find such an example. Any idea ? ...
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### Applications of Sard's theorem?

I'm writting an essay for University about differential topology. In particular, I'm studying the applications of Sard's theorem on differential topology. I have included the proof of Whitney's ...
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### Intuition regarding the $\sigma$ algebra of the past (stopping times)

Let $(\mathcal F_n)$ be a filtration and $\tau$ be a stopping time with values in $\mathbb N \cup \{\infty\}$. Let $\mathcal F_\infty$ be the sigma-algebra generated by $\cup_n \mathcal F_n$. ...
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### Showing that every finite measure on $(X,P(X))$ can be written as $\int_E f \,du$.

I want to show that every finite measure on $(X,P(X))$, where $P(X)$ denotes the power set of $X$ has the form $v(E) = \int_E f \,du$ for a non-negative measurable function $f$ and where $u$ is the ...
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### When do we have $\sup_{x}\int fdy=\int \sup_{x}fdy$?

When do we have $$\sup_{x}\int fdy=\int (\sup_{x}f)dy$$? Here, $f$ is a function of $x$ and $y$. Some say: Use The monotone convergence theorem, but I really don't understand that hint.
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### Topological structure & compactness of the space of probability measures

Let $(X,\tau)$ be a topological space and let $\mathcal P$ denote the set of all probability measures on $\mathscr B$, the Borel $\sigma$-algebra generated by the open subsets of $X$. In what follows, ...
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### Non-probabilistic analogue of the Second Borel-Cantelli lemma

The first Borel-Cantelli Lemma says that if we have events $E_i$ and $\sum_iP(E_i)<\infty$ then the probability infinitely many events occur is 0. The second is a partial converse saying if the ...
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### Showing $f\ge1$ a.e on $[a,b]$

Let $f$ be $\in L[a,b]$ Assume for any subinterval $I \subset [a,b]$ we have $\int_I f \geq |I|$ show that $f \geq 1$ a.e on $[a,b]$. I started with a proof by contradiction. Assume there exists a ...
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### If $xf \in L^2(\Bbb R)$, is $f \in L^1(\Bbb R)$?

I tried using Holder's inequality with $|f| = |f| \frac{(1+|x|)^{\frac{1}{2}}}{(1+|x|)^{\frac{1}{2}}}$, or variant but I can't seem to make it work. Any help is appreciated.
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### Erratum for Billingsley’s $\textit{Probability and Measure}$, Problem 33.9(c)

This is yet another verification request on a(n allegedly) false statement in Patrick Billingsley’s Probability and Measure textbook. I have been self-studying this excellent book for quite a while ...
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### A question on a proof of the existence of non-measurable sets

This is an excerpt of Rosenthal's book: "A First look at Rigorous Probability Theory" and contains an important theorem in the development of measure theory. While I generally understand the proof ...
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### Show that if $f$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$.
Let $A$ be a bounded measurable subset of $\mathbb{R}$. Show that if $f:A\rightarrow \mathbb{R}$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$. Choose real $c$. Since $f$ is ...
What is the definition of absolute continuity in whole $\mathbb{R}$. I know the definition on an interval $[a, b]$. I have a trouble with understanding the definition of absolute continuity in whole \$...