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Questions tagged [measure-theory]

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

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14 views

Derivative of Lebesgue integral with indicator functions

Suppose we have a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$. I want to take the derivative of $$ W\left(x\right)\equiv\int_{\Omega}\mathbf{1}\left\{ \omega\in R\left(x\right)\...
1 vote
1 answer
24 views

Prove independence of events given random variables are iid and have (not absolutely) continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_Y(y) := F_{Y_1}(y) = F_{Y_2}(y)...
2 votes
0 answers
30 views

Two sigma-algebras on $l^\infty$

Let $\mathcal{B}_w(l^\infty)$ denote the Borel $\sigma$-algebra generated by the weak topology on $l^\infty$. Let $Cyl(l^\infty)$ denote the cylindrical $\sigma$-algebra, that is, it is the $\sigma$-...
17 votes
1 answer
2k views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a nonnegative Radon measure $\mu$ on $\mathbb{R}^n$. Let $\mathcal{D}(\...
1 vote
1 answer
26 views

Let $A:=\{f\in C^1(\mathbb{R}): f, f'\in L^1(\mathbb{R}\}$. Then are the Schwartz functions dense in $A$ w.r.t. $\|f\|=\|f\|_1+\|f'\|_1$?

Let $A:=\{f\in C^1(\mathbb{R}): f, f'\in L^1(\mathbb{R}\}$. Then is it true that Schwartz functions are dense in $A$ with the norm $\|f\|=\|f\|_1+\|f'\|_1$.? My guess is the above statement shoud be ...
3 votes
2 answers
792 views

Prove independence of events given random variables are iid and have continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous. Denote common distribution a $$F(y) :=...
0 votes
0 answers
43 views

Understanding Measures in Perfectoid Spaces

I am currently studying perfectoid spaces and have encountered some difficulties understanding measures in this context. Background: Perfectoid spaces, as introduced by Peter Scholze, are a class of ...
0 votes
2 answers
53 views

For every measurable $E$ with $0<m(E)< \infty$, there is $[a,b]$ such that $m(E \cap [a,b]) > \frac{1}{2} m(E)$

Let $E$ be any Lebesgue measurable set in $\mathbb{R}$ with $0<m(E)< \infty$. I want to show that there exists a finite, nontrivial interval $[a,b]$ such that $m(E \cap [a,b]) > \frac{1}{2} m(...
1 vote
1 answer
53 views

Proof: If $f\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$, then $\{x\in X:|f(x)|>\|f\|_{\infty}\}$ is locally $\mu$-null.

I am self-studying Measure Theory by Donald Cohn. When proving this statement: If $f\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$, then $\{x\in X:|f(x)|>\|f\|_{\infty}\}$ is locally $\mu$-null. the ...
12 votes
4 answers
5k views

What is the motivation of Measure Theory when there is probability theory?

In my undergraduate studies, when probability was taught to me, it was taught to me starting from Probability Theory. However, when I go onto higher level of studies, probability gets taught using ...
1 vote
1 answer
147 views

Confusion with real numbers and random variables; Integration and Independence in Williams

In David Williams' Probability with Martingales, $\exists$ this exercise. Let $X_n$ be iid RVs with the same continuous distribution function. Let $E_1 = \Omega$ and for $n \geq 2, E_n = (X_n > ...
1 vote
1 answer
96 views

Proof in measure theory

I have found excellent proof about changing variables in polar coordinate This is the problem Let $S^{n-1}=\left\{x \in R^2:|x|=1\right\}$ and for any Borel set $E \in S^{n-1}$ set $E *=\{r \theta: 0&...
3 votes
2 answers
49 views

Proving that restriction of $\sigma$-algebra is a $\sigma$-algebra

Let $X$ be a set and $\Sigma$ be a $\sigma$-algebra of $X$. Show that $\Sigma \cap A = \{E \cap A: E \in \Sigma \}$ is a $\sigma$-algebra of $X$ My attempt actually showed that this is impossible: ...
1 vote
1 answer
70 views

Understanding the meaning of measurability [duplicate]

I am trying to get an intuitive understanding of what it means for a function $X$ to be $\mathcal{H}$-measurable, specifically to understand why we require the conditional expectation $\mathbb{E}[X|\...
3 votes
1 answer
55 views

Radon Nikodym derivative and distribution function

Let $\mathfrak{B}$ be the Borel $\sigma$-algebra over $\mathbb{R}$ and $\beta$ the Borel-Lebesgue measure over $\mathfrak{B}$. Let $\mu$ be a Borel measure over $\mathbb{R}$ s.t. the distribution ...
2 votes
1 answer
40 views

Is every collection of discrete random variables a function of independent random variables?

Let $\Omega = \{0,1\}^n$, and let $(\Omega,2^\Omega,\mu)$ be a probability space. Does there exist $\Omega' = \{0,1\}^N$, product probability measure $\mu' = \mu_1 \otimes \dots \otimes \mu_n$ on $(\...
2 votes
1 answer
82 views

Stokes theorem for currents on manifolds with corners

Let $M\subset\mathbb R^N$ be a compact oriented $n$-(sub)manifold with corners and $\omega$ be an $(n-1)$-form on it. The usual statement of Stokes theorem $$\int_M d\omega=\int_{\partial M}\omega$$ ...
0 votes
0 answers
39 views

Integration with respect to finite Radon measure

Let $ u \in BV( \mathbb{R}^N). $ We know that $$ \int_{\mathbb{R}^N} \left|Du\right| = \sup\left\{ \int_{\mathbb{R}^N} u\ div \varphi\ dx,\ \varphi \in C_c^1( \mathbb{R}^N, \mathbb{R}^N),\ \left|\...
3 votes
1 answer
57 views

If $E \in \mathcal{A}$ satisfies $\mu(E)>0$, then there exists $F \subset E, F \in \mathcal{A}$ with $0 < \mu(F) < \infty$

Suppose $(X, \mathcal{A}, \mu)$ is a $\sigma$-finite measure space. If $E \in \mathcal{A}$ satisfies $\mu(E)>0$, then there exists $F \subset E, F \in \mathcal{A}$ with $0 < \mu(F) < \infty$ ...
1 vote
1 answer
303 views

$\sigma$-finite measure $L^p$ space is isometric to a finite measure $L^p$ space

I have a measure space $(X,M,\mu)$ where $\mu$ is $\sigma$-finite. Then there exist a finite measure $\lambda$ s.t. the space $L^p(\mu)$ is isometric to $L^p(\lambda)$. First I proved that there ...
3 votes
2 answers
193 views

Help in understanding this, convergence in measure.

Let $(X,A,\mu)$ be a measurable space and let $ g,f_1,f_2,\ldots \in L^1(\mu)$, $g\geq 0$ and $f:X\to \mathbb C$ be measurable. Suppose that $f_n$ converges to $f$ in measure ($\mu$) and that $|f_n|\...
0 votes
0 answers
16 views

Showing that $σ(X_1, X_2, . . .) = σ({{X_1 ≤ x_1, . . . , X_r ≤ x_r} | x_1, . . . , x_r ∈ \mathbb{R}, r ∈ \mathbb{N}}).$

Let $\left(X_{n}\right)_{n} $ be a sequence of real-valued random variables. Show that $ \sigma\left(X_{1}, X_{2}, \ldots\right)=\sigma\left(\left\{\left\{X_{1} \leq x_{1}, \ldots, X_{r} \leq x_{r}\...
2 votes
1 answer
150 views

Inner measure doesn't care whether finite or countable

I was reading this answer, to try understanding why the inner measure defined as the $\mu_*(U)=\sup\{\mu(A):A\subset U\text{ is measurable}\}$ is not really used. I copied the answer here : Tao talks ...
1 vote
0 answers
79 views

Is there a measure that produces given values (probabilities or cardinals) for sets $A_1,\dots, A_n$ and all their intersections $A_i\cap A_j, ... $?

Assume that values (e.g., probabilities or cardinals) of a measure on a finite set $\Omega$ are given for sets $A_1,\dots, A_n$ and all of their intersections $A_i, A_i\cap A_j, A_i\cap A_j\cap A_k, ....
37 votes
2 answers
8k views

The measurability of convex sets

How to prove the measurability of convex sets in $R^n$? I have seen a proof, but too long and not very intuitive. If you have seen any, please post it here.
2 votes
2 answers
2k views

David Williams "Probability with Martingales" Exercise 4.1

Let me preface this by saying that basically the same question has been asked before on MathStackExchange. However, there is one small detail in an exercise that I cannot reconcile. The following ...
1 vote
1 answer
24 views

Prove that $\lim_{k \to \infty} \int_{\mathbb{R}} g_k fdm_1 = f(0)$

Suppose that $g$ is a non negative Borel measurable function on $\mathbb{R}$ with $\int_{\mathbb{R}} g dm_1 = 1$ where $m_1$ denotes the one-dimensional Lebesgue measure. For $k \in \mathbb{N}$ set $...
-2 votes
0 answers
64 views

Can you solve $\int_0^1\!\sqrt{x^4+4x^2+3}\,\mathrm{d}x$ using Feynman's differentation under the integral sign trick? [closed]

My question is what stands in the title. Can you solve $\displaystyle\int_0^1\sqrt{x^4+4x^2+3}\,\mathrm{d}x$ using Feynman's differentation under the integral sign trick?
0 votes
0 answers
20 views

Growth of uniformly convex function: Elementary proof

It is known that if $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is uniformly convex, i.e. there is $\mu > 0$ such that $$ f(tx+(1-t)y) \leq tf(x) + (1-t)f(y) -\mu t \lVert y-x \rVert^2~~~ \forall (x, y)...
4 votes
1 answer
166 views

Measure of projection of a set is zero

Suppose $S$ is measurable in $[0,1]^2$ and the orthogonal projection of $S$ onto the $x$ and $y$ axes has measure zero. Can the measure of the orthogonal projection of $S$ onto the line $y=x$ be ...
8 votes
3 answers
2k views

Is it true that |X| vanishes at infinity, then X is integrable?

Suppose $X$ is a random variable on $(\Omega,\mathscr{F},\mathbb{P})$. If there exists $M>0$, such that for all $\lambda>0$, we have $\mathbb{P}[|X|>\lambda]\le\frac{M}{\lambda}$, then is it ...
1 vote
0 answers
31 views

Layer cake representation and function with compact support

My question is related to the posts here and here, but my setup is slightly different. For a real-valued random variable $X$, and a function $\varphi: \mathbb{R} \to \mathbb{R}$ that has support in ...
0 votes
0 answers
48 views

Why do we need the "Size $\delta$ Approximations" to Hausdorff measures?

The sources I've looked at define the Hausdorff measure in two stages. They first define, for dimension $s$ and some dimension-dependent constant $\alpha(s)$: $$ H_\delta^s(A) =\inf \left\lbrace \sum_{...
1 vote
2 answers
47 views

Why do we require intervals in Jensen's inequality?

In my notes I have where $I$ is the interval on which $X$ takes values. I am struggling to see why do we require for our random variable to be contained in an interval. I thought looking at the ...
0 votes
0 answers
29 views

Show that there exists a function F which is a measure of intervals in the real line

I've just started self-studying measure theory and the book I'm using has the following exercise. Let $J$ be an interval in $\mathbb{R}$ and $\mathcal{S}_J = \{S \subseteq J\ | S\ \text{is bounded} \}...
5 votes
2 answers
1k views

$f_x$ is Borel measurable and $f^y$ is continuous then $f$ is Borel measurable

I have to prove the following: Let $f: \mathbb{R^2}\to \mathbb{R}$ such that $f_x:y\to f(x,y)$ is Borel measurable for all $x\in\mathbb{R}$ and that $f^y:x\to f(x,y)$ is continuous for all $y\in\...
2 votes
1 answer
25 views

Finite linear combination of characteristic functions measurable implies each set is measurable?

Question : Let $X$ be a set and $\mathcal{F}$ be a sigma algebra for $X$. Let $E_1$, ...., $E_k$ be subsets of $X$. If $g(x)$ = $\sum \alpha_j \chi_{E_j}$ is measurable (where $\alpha_j$'s are ...
1 vote
0 answers
38 views

Show that if $f$ is a measurable complex-valued function on $(X,\mathscr{A})$, then $|f|$ is also measurable.

I need to show If $f$ is a measurable complex-valued function on $(X,\mathscr{A})$, then $|f|$ is also measurable. I tried it myself, but don't know if my work is correct or not? Could someone ...
0 votes
1 answer
49 views

"Leibniz's rule" for $t\in\mathbb{R}^n$

I am looking for a reference giving a measure-theoretic proof of a claim from the German Wikipedia. I have searched the references given on that site, as well as the English speaking Wikipedia and all ...
1 vote
0 answers
52 views

$A \subset [0,1]$. $P_a$ is a parabola tangent to OX in $(a,0)$. $B= \bigcup_a P_a \cap [0,a] \times R$. Show: $\lambda_2(B)=0 \iff \lambda_1(A)=0$

$A \subset [0,1]$. $\forall_{a \in A}$ we name as $P_a$ a parabola that is tangent to $OX$ in point $(a,0)$. $B = \bigcup_a P_a \cap [0,a] \times \mathbb{R}$ Show that: $$\lambda_2(B) = 0 \iff \...
0 votes
1 answer
23 views

If $\mathcal{A'}$ is an algebra then $\mathcal{A}=\{A \cap E : A \in \mathcal{A'}\}$ is an algebra on $E.$

Let $\mathcal{A'}$ be an algebra on a non-empty set $X$ and $E \subset X$, then $\mathcal{A}=\{A \cap E : A \in \mathcal{A'}\}$ is an algebra on $E.$ I was able to show that; $\mathcal{A}$ is closed ...
1 vote
1 answer
79 views

The Medians of Lipschitz Functions on $(X,d,\mu)$ (Existence and Uniqueness)

Let $\varphi:(X,d,\mu)\to \Bbb R$ be a Lipschitz function, where $\mu$ is a probability measure on the metric space $(X,d)$. The median $m_\varphi$ of $\varphi$ is defined as the real number such that ...
2 votes
1 answer
50 views

Equivalence of definitions of "standard Borel space"

I met the following definition of standard Borel spaces in Durrett's probability theory book (slightly rephrased): $(S,\mathcal{S})$ is said to be standard Borel if it is isomorphic (as a measurable ...
0 votes
0 answers
30 views

Relation between $2$-dimensional and $1$-dimensional Lebesgue measure

Let $Q\subset\mathbb{R}^n$ be a compact set (if you want say $Q=[0,1]^n$). My question is: If $S\subset Q\times Q\subset\mathbb{R}^{2d}$ is a measurable set with $|(Q\times Q)\setminus S|<\epsilon$,...
1 vote
0 answers
36 views

Lebesgue inner measure asymmetry

In Tao's book there is a sentence: "Here, there is an asymmetry (which ultimately arises from the fact that elementary measure is subadditive rather than superadditive): one does not gain any ...
0 votes
0 answers
48 views

Prove that $\int_X \limsup f_n \, d\mu \geq \limsup \int_X f_n \, d\mu$

Let $(X, \mathcal{A}, \mu)$ be a measure space. Suppose that $f_n$ is a sequence of measurable functions such that there is a non-negative measurable function $F$ satisyfing $$\int_X F \, d \mu < \...
1 vote
0 answers
36 views

Distribution induced by a Radon measure

Let $\Omega \subset \mathbb{R}^N$ be open, and consider a distribution $T\in \mathcal{D}’(\Omega)$. I should prove the following statement. The distribution $T$ is a linear combination of Radon ...
0 votes
1 answer
2k views

Question about total variation, positive variation and negative variation

I have a question about the following problem: Let $f:[a,b] \to \mathbb{R}$ be a function of bounded variation with $f(a) = 0$ and $f_1$, $f_2$ be two increasing function such that $f_1(a) = 0$ and ...
0 votes
0 answers
32 views

doubt about proof in folland book

I am having hard time to understand a roof in the book $1$ what does it mean to be a Borel measure on $S^{n-1}$? $2$ how does he obtained $=\rho \times \sigma((a, b] \times E) \text {. }$?
3 votes
1 answer
50 views

$\left |\int _{X} log(|f|)d\mu \right | < \infty$ if $\int_{X} |f| d\mu <\infty $?

Is the following true?: Assume that $\mu$ is any positive measure with $\mu (X)=1$ and that, if $E_{0}=\left \{ x\in X :|f(x)|=0 \right \}$, we have $\mu (E_{0})=0$. Then $$\left |\int _{X} log(|f|)d\...

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