# Tagged Questions

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

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### Conditions for a function to lie in $L^p(\mathbb{R})$

Let $(X, \mathfrak{M})$ be a measurable space. What are some sufficient and necessary conditions for a function $f : X \to \mathbb{R}$ to lie in $L^p(\mathbb{R})$ for $p \in [1,\infty]$? Is true ...
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### Necessary and sufficient conditions (1) rv to density function (2) distribution to rv

(1) Let $(\Omega,\mathcal{F},P)$ be a probability measure space and $X:\Omega \rightarrow \mathbb{R}$ a random variable. Let $P_X,~F_X$ denote the probability measure, pdf induced by $X$, ...
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### Sufficient and necessary conditions for representation of a ordered structure with a binary operation.

Given a structure $\mathcal{A} = (A, \succsim, \sqcup)$, where $A$ is a non-empty set, $\succsim$ is a weak order, $\sqcup$ is a binary operation on $A$, let $\mu$ be an order-preserving mapping from ...
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### Integral of a measurable function

I do not know what should i keep as title for this question... Question goes like this.. Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a measurable function. If $\int_{-\infty}^{\infty}f(x)dx=1$ prove ...
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### Is there a probability measure on the Cantor set?

I know that the Lebesgue measure of the Cantor set is $0$. Is there a finite positive regular measure on the Cantor set?
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### Lebesgue Measure of $A=\left \{ (x,0) : x \in [0,1]\right \} \subset \mathbb{R}^2$

Let $A=\left \{ (x,0) : x \in [0,1]\right \} \subset \mathbb{R}^2$ and $m_2$ Lebesgue Measure of $\mathbb{R}^2$. I want to determine $m_2(A)$. So. I know that Lebesgue Measure of interval is b-a. And ...
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### Several questions about Riesz–Markov–Kakutani representation theorem

This is a list of questions about Riesz–Markov–Kakutani representation theorem . 1)If $f\in L^1(\mu)$, is it true that $\phi(f)=\int_Xfd\mu$, where $\mu$ is given by the theorem? I am quite sure it ...
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### Are all sets with finite measure measurable?

In my textbook, it says: "Let E be any set with m*(E) < $\infty$. Then E is measurable if and only if there exists a measurable set B with m(B) = m*(E)." There always exists a measurable set of ...
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### Sigma Algebra - Partition

Let $\Omega = \{1, 2, . . . , 7\}$ and let $A = \{\{1, 2, 3, 7\}, \{2, 3, 4, 5, 6\}\}$. Find $P(A)$. P is for Partition. I got $P(A) = \{\{1,4,7\}, \{2,3\}, \{5,6\}\}$ If this is wrong, can you ...
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### Simple random walk on $\mathbb Z^d$ and its generator

I'm still trying to figure out definitions and properties of random walks on $\mathbb Z^d$. My goal is to work up to understanding some large deviation principles for the local times of such random ...
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### Distribution function derivative bounds give bounds on associated measures? Billingsley theorem 31.4 proof.

I am working through Billingsley, Probability & Measure. Struggling with the proof of theorem 31.4: Suppose $u(a,b) = F(b) - F(a)$ and that $F'$ exists throughout a Borel set $A$. If $F' ≤ c$ ...
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### Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A$ such that $||x^{(N)}||\to \infty$ ...
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### For $\int f < \infty$, the measure of the set of points where $f=\infty$ is zero.

I fear this question was already discussed here, but I was not able to find it. Please remove if it is a duplicate. Prove: For a function $f\geq 0$, if $\int f < \infty$, then the measure of ...
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### Proving the monotonicity of a countably additive set function on a $\sigma$-algebra [on hold]

Let m be a set function defined for all sets in a $\sigma$-Algebra $\scr A$ ; Assume that m is countably additive over countable disjoint collections of sets in A , with values in $[0,\infty ]$ ...
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### Differentiable function has measurable derivative?

Let $f:[0,T] \to \mathbb{R}$ be a differentiable function. Is it true that $f'$ is measurable? If so, is this also true if $f$ is differentiable almost everywhere? Sorry for lack of effort but I ...
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### Show that f is measurable.

Let $a > 0, b \geq 0$ and the function $f: \mathbb{R} \to \mathbb{R}$ $$f(x) = \left\{\begin{matrix} 1, & |x| \leq a \\ b & |x| > a \end{matrix}\right.$$ show that it is measurable. ...
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### $\sigma$-algebra generated by weak topology in Hilbert Space

In general, if we have $H$ Hilbert space, and equipped with the weak topology, say $\tau^\ast$, is $\sigma(\tau^*)=\mathcal{B}$?, where $\mathcal{B}$ is the usual Borel $\sigma$-algebra I suspect it ...
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### Measure $m=\mu$ if $\int fdm=\int fd\mu$

Suppose $X$ is a locally compact Hausdorff space, $m,\mu$ are two Borel measures, if for any $f\in C_c(X)$, $\int fdm=\int fd\mu$, is it true that $m=\mu$?
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### How can I show that one of $m(A)$ or $m(\Bbb{R}\setminus A)$ is zero?

Let $A \subseteq \Bbb{R}$ be Borel measurable, and $T$ a dense subset of $\Bbb{R}$. Suppose for every $t \in T$ that $$m((A+t)\setminus A)=0,$$ where $m$ is the Lebesgue measure. Then I want to show ...
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### Defining a measure by positive functional

In big Rudin's book, it constructs the Lebesgue measure by first defining a positive functional, and then using Riesz representation theorem. It arises me to think that if every measure can be ...
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### If K=P(X) , then λ is a pre outer measure if and only if it is an outer measure. [on hold]

If K=P(X) where K is an algebra, then λ is a pre outer measure if and only if it is an outer measure. Is it enough to prove that all sets in K are measurable? Any suggestions.
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### n-1 dimensionnal Hausdorff measure and codimension 1 measure

I've been told that on a n-dimensionnal Riemannian manifold, the Hausdorff measure of dimension n-1 and the codimension 1 measure $v_{-1}$ (defined below) are mutually absolutely continuous. I've ...