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

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5
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1answer
304 views

how to show that integral depending on a parameter are continuously differentiable

I'm trying to solve this exercise Let $f:[0,1]\to \mathbb{R} \space$ an integrable function, show: $$g(z):=\int_{0}^1\frac{f(x)}{x-z}dx$$ is a continuous differentiable function on $\mathbb R ...
0
votes
1answer
59 views

Proof of measurability of a function.

I am currently reading through my notes and I have some trouble to understand this solution: Let $(X, \mathbb X)$ be a general measurable space. Then, as a partial exercise: Let $(A, \mathbb A)$ ...
0
votes
1answer
93 views

Understanding this inner product

I want to find out under which conditions on $w$, we have that $$\langle f,g \rangle :=\int_0^1 f(x)\bar{g}(x)w(x) dx $$ a dot product?, where $f,g \in C([0,1],\mathbb{C})$ and $w \in ...
2
votes
1answer
88 views

Measure and Probability

Can someone tell me that how did the idea to relate measure and probability come?(What's the conceptual history of measure and probability?)
1
vote
1answer
83 views

Stopping time and filtrations

I have a definition problem. I know that a filtration on a probability space is an increasing sequence of $\sigma$-algebras. I was now thinking on the fact that constant times are stopping times. I've ...
1
vote
1answer
76 views

Regular conditional probability living on sections

Let $X$ and $\bar X$ be standard Borel spaces, let $A\subseteq X\times \bar X$ be an analytic subset of the product space and let $P$ be a probability measure such that $P(A) = 1$. Does there exists a ...
2
votes
1answer
293 views

Hardy-Littlewood maximal function weak type estimate

Show that if $f\in L^1(\mathbb{R}^d)$ and $E\subset \mathbb{R}^d$ has finite measure, then for any $0<q<1$, $$\int_E |f^{*}(x)|^q dx\leq C_q|E|^{1-q}||f||_{L^1(\mathbb{R}^d)}^{q}$$ where $C_q$ ...
5
votes
1answer
63 views

Show that $\mathfrak{Z}$ is a semi ring

Consider measurable spaces $(\Omega_t,\mathcal{A}_t), t\in T$ ($T$ is any index set). With $\mathcal{E}(T)$ we the set of all finite, not-empty subsets of $T$. Show that $$ ...
0
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1answer
245 views

When a semi-ring which contains a union?

In what follows $\bigsqcup$ denotes disjoint union. I have marked the point where I don't understand the reasoning with (WHY). In the proof(Measures integrals and martingales starts p 39)(step 2): ...
1
vote
1answer
53 views

One implication (on Measure)

Please be noted that charges are finitely additive measures and measure are countably additive ones. Theorem 2.1. Let $\mu$ be a charge on a Boolean algebra $B$. Each of the following conditions ...
3
votes
2answers
143 views

How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$

Let $f:\mathbb{R}\to\mathbb{R}$ be Borel-measurable and Lebesgue-integrable over $I:=(0,1)$. Further, let $\;\;\;\;\;\;\;\;\;\;g : I\to \mathbb{R}\;,\;\;\; \displaystyle x ...
0
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1answer
72 views

Change of variable with measures other than the Lebesgue measure.

I ask my question with a specific example in mind. Consider the integral \begin{align} I_k=\int_{\mathbb R}(2\cos(x))^k~d\mu(x),&&k\in\mathbb N\tag{1} \end{align} where ...
0
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1answer
51 views

the Infinite sum of a sequence of function can be considered as a function itself?

Let $f_1$,$f_2$, $...$ be a sequence of non-negative measurable functions on a measure space $(X,\nu)$. Suppose that $$\sum_{n=1}^{\infty}\int_X {f_n}(x) d\nu \lt \infty $$ Prove that this implies ...
3
votes
0answers
112 views

Tricky weak-* convergence question

Let $K$ be a compact (let's say, $K=[0,1]$ to be concrete) and let $\mu_n$ be a sequence of Radon measures converging weakly-* to another Radon measure $\mu$ on $K$. Let $h : K \rightarrow \mathbb{R}$ ...
6
votes
2answers
285 views

How to prove $\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}$

I'm asked to prove $$\displaystyle\int_0^\infty e^{-x}\frac{\sin^2 x}{x}\text{ dx}=\frac{\text{log }5}{4}\tag{$\ast$}$$ by integration of $e^{-x}\text{sin}(2xy)$ over an suitable measurable ...
3
votes
1answer
265 views

Lebesgue measure of a set containing numbers without zero in their decimal expansion.

Let A ⊆ [0,1] be the set of real numbers which do not have zero in their decimal expansion. Can someone give me a hint that helps me compute the Lebesgue measure of this subset? I think that I have ...
0
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1answer
35 views

Invariant measure on Graff(2,1)

What is the invariant measure on Graff$(2,1)$ the set of all straight lines in $\mathbb R^2$? I tried looking at it this way: One is aware that Graff$(2,1)$ can be identified with the canonical ...
0
votes
4answers
118 views

Boundness & integrability

Let's take a function $f\in L^1$ Does it follows that f is also bounded? Couldn't it be unbounded on zero-sets? I'm working in probability theory (finite measure spaces) Thanks for any help
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vote
2answers
295 views

Proving the Exterior (Outer) Measure of Rectangle is Equal to Volume

I'm having trouble understanding one step of Stein and Shakarchi's proof that the exterior measure of a rectangle is equal to its volume. The proof I reference is part of Example 4 in section 1.2 of ...
1
vote
1answer
55 views

Pre-image of conditional expectations

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, $S:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $\mathcal F_0\subseteq\mathcal F$ and $H\subseteq\mathbb R^d$ a Borel ...
3
votes
1answer
93 views

Measures $\psi$ is the smallest under all other measures with this property

Let $\alpha$ and $\beta$ be two finite measures, in particualar $\alpha(A\cup B)=\alpha(A)+\alpha(B)$ if $A\cap B=\varnothing$ (and also for $\beta$) and if $A_1,A_2,\ldots$ are a decreasing sequence ...
1
vote
2answers
75 views

Show: $\mu(A\Delta B)=0\implies \mu(A)=\mu(B)$

Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $A,B\in\mathfrak{A}$. Show: $\mu(A\Delta B)=0\implies\mu(A)=\mu(B)$. Hey, I've tried to prove that, unfortunately without success ...
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0answers
31 views

Simple function for pairs disjoints sets

Prove that every simple function we can write as $$f(x) = \sum_{k=1}^{n} a_k 1_{A_k}(x)$$ where $A_k$ are pairs disjoints sets. My attempt: Let $B_1,...,B_n$ be any sets. We define $A_n$ sets as: ...
0
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1answer
68 views

what are measurable spaces on the real line?

I've came across this article about the dominated convergence theorem , but since i didn't take a course on measure theory , i have some problems understanding the language of the previously or other ...
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0answers
33 views

Prove this RV converges in probability

Let $X$ be a Possion random variable with parameter $\lambda \theta$. Show that $X-\lambda \theta \over \lambda$ converges to zero in probability, as $\lambda \rightarrow \infty$, in other words ...
0
votes
2answers
41 views

Determine polynomials with $n$-variables

Here is a funny problem arise from harmonic analysis: Let $E$ be a measurable subset of $\mathbb R^n$ with $m(E)>0$, where $m$ is the usual Lebesgue measure on $\mathbb R^n$. In practice, $E$ ...
5
votes
3answers
596 views

Dirac delta of a function with zero derivative

It is known that: $$\int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}$$ Where $x_i$ are the roots of $g(x)$. My question is, what happens when $g'(x_i)$ is ...
1
vote
1answer
69 views

A metric defined on the space of all positive finite measures.

Let $\mathbb M$ be the space of all positive finite measures on a measurable space $(X,\mathcal{M})$. $$(\lambda,\nu) \mapsto d(\lambda,\nu) \equiv 2\sup_{E \in \mathcal M} |\lambda(E) - \nu(E)|$$ is ...
2
votes
0answers
88 views

Strategies for swapping the order of integration with dependent bounds

What are the general strategies for swapping the order of integration given dependent bounds? Specifically, in $\mathbb{R}^2$, Fubini's theorem allows us the following $$ \int_{a}^b\int_{c}^d ...
0
votes
1answer
303 views

Proof of uniqueness of the extension in Kolmogorov extension theorem

Statement of the theorem. The proof is mainly focused on showing that the candidate probability measure defined on the algebra of sets is $\sigma $-additive. At the end, the Hahn-Kolmogorov theorem ...
4
votes
3answers
106 views

Beyond Calculus?? Integral Convergence using Measure Theory & Real Analysis

$$ \mbox{Does}\quad \int_{\pi}^{\infty} {{\rm d}x \over x^{2}\sin^{2/3}\left(x\right)}\quad \mbox{diverge ?} $$ Is this problem suitable for a calculus class ?. I'm not sure exactly how to solve but ...
2
votes
1answer
160 views

Show $\psi$ and $\Delta$ are identifiable

Let $X_1$,...,$X_m$ be i.i.d. F, $Y_1$,...,$Y_n$ be i.i.d. G, where model {(F,G)} is described by $\hspace{20mm}$ $\psi$($X_1$) = $Z_1$, $\psi$($Y_1$)=$Z'_1$ + $\Delta$, where $\psi$ is an unknown ...
1
vote
1answer
122 views

munkres analysis integration question

Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2 \to \mathbb{R}$ be defined by setting $f(x,y)=0$ if $y \neq x$, and $f(x,y) = 1$ if $y=x$. Show that $f$ is integrable over $[0,1]^2$.
0
votes
1answer
88 views

Prove these random variables are independent

Let $X_0,X_1,X_2,...$ be independent random variables with $$P(X_n =1)=P(X_n =-1)={1\over 2},\forall n$$ Let $$Z_n = \prod_{i=0}^n X_i .$$ Show that $Z_1,Z_2,...$ are independent. An intuition is ...
1
vote
0answers
82 views

How come these two sums are the same? (Quiet terse, but please help..!)

I'm struggling this for about two days... Below describes the situation. I'm trying to prove the finite additivity of Lebesgue measure. Define $\mathscr{B}=\{\prod_{i=1}^n [a_i,b_i) : ...
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0answers
140 views

Question about a part of the extension theorem proof

My question is how can we justify interchanging the union to get $(*)$. I think I did it see edit 3 or answer. In the proof(Measures integrals and martingales starts p 39)(step 2): the following set ...
1
vote
1answer
65 views

How do i do this process *precisely*?

Let $[a,b)\times [c,d)$ be a rectangle $R$ in $\mathbb{R}^2$. Let $\{[u_k,v_k)\times [p_k,q_k)\}_{1≦k≦n}$ be a mutually disjoint finite sequence whose union is $R$. Then we can decompose this into ...
1
vote
2answers
92 views

Radon Nikodym derivative of measures on $\mathcal{B}_{(0,\infty)}$.

(a) Show that there is at most one measure $\nu$ on $\mathcal{B}_{(0,\infty)}$ which satisfies the following conditions: $\nu((1,e]) = 1$ $\nu(cA) = \nu(A)$ for every $c>0$ and $A \in ...
1
vote
2answers
93 views

Measure of big discontinuities

Let $D\subset\left[ 0,1\right] $ be a dense set, and $\mu$ Lebesgue measure on $\left[ 0,1\right] .$ Suppose $f:\left[ 0,1\right] \rightarrow\left[ 0,1\right] $ is continuous at each point in ...
1
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1answer
115 views

Help proving that a measure is absolutely continuous with respect with respect to another measure

Suppose that $E$ is a locally compact and separable metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In ...
2
votes
1answer
287 views

Integration, Lebesgue and counting measure

Could you help me with the following exercise? Consider $X=Y=[0,1]$ with Lebesgue measure $m$ on $X$ and counting measure $\omega$ on $Y$. Let $f:X \times Y \rightarrow \mathbb{R}$ and $f(x,y)= ...
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1answer
41 views

Using approximation of a function $f \in L^1 \cap L^\infty$.

Let $f \in L^1(m) \cap L^\infty(m)$. Define a functions $\varphi$ on $\mathcal{R}$ by letting $\varphi(x) := \int f(x-t)f(t)dm(t)$. (a) Prove that the integral defining $\varphi$ is well defined. ...
0
votes
1answer
28 views

How do i prove that the set of all $Q$-box is a semi ring?

I'm trying to prove the uniqueness of the lesbesgue measure. Typically, there are two distinct ways to construct the lebesgue measure. One is via outer measure and one is via completion of borel ...
1
vote
1answer
359 views

Is the limit of an uniformly convergent sequence of integrable functions (with convergent integrands) integrable?

My question is the following: Consider a sequence of Lebesgue integrable functions $f_{n}$ (over R) that converges uniformly. Assume furthermore that the integrands $\int f_{n}$ converge to some ...
1
vote
1answer
352 views

Version of the Vitali Covering Lemma

Let $\mathcal{B}$ be a collection (not necessarily finite) of open balls in $\mathbb{R}^d$ with the property $| \cup_{B\in \mathcal{B}} B|<\infty$. Prove that there exists a countable subcollection ...
0
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1answer
66 views

Measurability of a regular set

This is a problem from an old comp exam. Let $\mu^*$ be an outer measure on $X$, with $\mu^*(X) < \infty$. Suppose $E \subset X$ is regular, i.e. there is a $\mu^*$-measurable set $A \supset E$ ...
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votes
2answers
91 views

The measure of the closure of borel measurable sets.

Given lebesgue measure $m$ on $\mathbb R^n$, I am interested in whether $m(E) = m(\overline E)$ for all $E$ and why. In general, if $E$ is bounded, then by observing the $\overline E \subset U$ ...
2
votes
1answer
76 views

Reference Request: Semi-Rings and Rings (System of Sets, not Algebraic Structures)

I studied Probability Theory (from a Measure Theory viewpoint) using only Sigma-Algebras. Recently, I got a book about measure theory that starts from Semi-Rings, but it's presentation is too ...
5
votes
1answer
111 views

If $0\leq f_n$ and $f_n\rightarrow f$ a.e and $\lim\int_Xf_n=\int_X f$, is it true that $\lim\int_Ef_n=\int_E f$ for all $E\in\mathcal{M}$.

If $0\leq f_n$ and $f_n\rightarrow f$ a.e and $\lim\int_Xf_n=\int_X f$, p,rove or disprove that $\lim\int_Ef_n=\int_E f$ for all $E\in\mathcal{M}$. I think it is true. It is easy to see ...
1
vote
0answers
90 views

Largest $\sigma$-algebra on which $\mu$ is uniquely extendable exists?

Let $\mu$ be a $\sigma$-finite pre-measure a semi-ring and let ${\cal A}$ denote the $\sigma$-algebra of Caratheodory-measurable sets. I understand that for $D\notin {\cal A}$ one can construct an ...