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

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Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg\, d\mu :\|g\|_\infty \leq 1\}$

Let $\mu $ be a $\sigma $-finite measure. Show that for any $f\in L_p(\mu)$, $\|f\|_1=\sup\{ \int fg \, d\mu :\|g\|_\infty \leq 1\}$ I know that Holders inequality implies $\int fg \, d\mu \leq ...
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1answer
18 views

Exercise 10.J of The elements of integration and Lebesgue measure Bartle's book

The part of the problem is the next. Let (X,X,$\mu$) be the measure space on the natural numbers X=$\mathbb{N}$ with the counting measure defined on all subsets of X=$\mathbb{N}$. Let (Y,Y,$\nu$) be ...
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1answer
22 views

Integral as a member of the closure of the convex hull of the integrand

Suppose that $X$ is compact and metric and let $g:X\to\mathbb R$ be a Borel map. Let $\mu$ be a Borel probability measure on $X$. Then it seems that $\int_Xgd\mu$ is a member of the closure of the ...
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2answers
45 views

Product measure problem

I wonder if you can help me out with this problem that I'm trying to understand (from an old exam, not homework): Let $(E,\mathcal{P}(E))\,$ (where $E= \lbrace 0,1 \rbrace )$ be a measurable space, ...
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1answer
94 views

Measure and set theory.

I have read that if we assume the continuum hypothesis then it can be proved or concluded tha there exist a set function μ that has the three following properties: μ(A) is defined for each set A of ...
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0answers
30 views

Linearity of Lebesgue measure

Suppose $\mu$ is the Lebesgue measure defined on $\Bbb R^k$, I want to show that $\mu$ has some kind of linearity, which seems intuitively correct: Suppose $A$ is a linear transformation on $\Bbb ...
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1answer
51 views

Hellinger integral properties - proof of equivalence for infinite product measures

I'm trying to prove that: Let $(\mu_k)_{k=1}^{\infty}$ and $(\nu_k)_{k=1}^{\infty}$ be sequences of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Consider the product measures on ...
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0answers
19 views

Integrability of a measurable function

Hi everyone: Suppose $(E_{n})$ is an increasing sequence of sets in $\mathbb{R}^{p}$ $(p\geq2)$ such that $\bigcup_{n}E_{n}= B$, a ball in $\mathbb{R}^{p}$. Suppose also that $f$ is a measurable ...
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0answers
54 views

Carother's “certainly” proof about measurable sets

Carother's Real Analysis text has the following Theorem. Can someone check if my proof is correct? $(i \Rightarrow ii)$ Let $E$ be a measurable. Let $I_k$ be open intervals, such that $$m^*(E) ...
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1answer
92 views

Why Are Some Sets Not Measurable?

I'm trying to understand why you can't evaluate a measure on generic sets (the ones in Banach-Tarski construction). That is, I want to know why when considering $m(X)$, we have to restrict our ...
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1answer
23 views

Definition of Hausdorff Measure: example question

I am studying the Hausdorff measure and dimension, but I am struggling to understand the reason that the $n$-dimensional Hausdorff measure is zero for a set with Hausdorff dimension $<n$. The ...
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0answers
29 views

Visualization of Fubini's Theorem

I understand that Fubini's Theorem is vital to evaluating double and triple integrals (via the equivalence of iterated integrals) especially in elementary multivariable calculus, and I know that it ...
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1answer
31 views

Fatou Lemma: Why is $\lim\inf f_n = 0$ where $f_n = \chi_{[n,n+1]}$

In this wildly popular post, there is a claim: I like to remember this by example; specifically let $f_n = \chi_{[n,n+1]}$. Then $\lim \inf f_n = 0$, and $\lim \inf \int f_n = 1$. So $f_n = ...
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1answer
68 views

Compute the Lebesgue integral $\int_0^{\infty} \frac{x}{e^x -1}dx$.

Compute the Lebesgue integral $\int_0^{\infty} \frac{x}{e^x -1}dx$. I think I need to use the Dominated Convergence Theorem or the Beppo Levi Theorem to show this, but I don't really know what I ...
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1answer
34 views

Induce measure between topological spaces.

Let $X$ and $Y$ be topological spaces, suppose that the function $f:X\rightarrow Y$ is a continuous surjection. Let $\mu$ be a regular measure on $X$, and $M_X$ the set of $\mu$-measurable sets in ...
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0answers
13 views

estimate on the sum of Rademacher functions

Let $(r_n)_{n\in\mathbb{N}}$ be Rademacher functions defined on $[0,1]$. See https://en.wikipedia.org/wiki/Rademacher_system for the definition of Rademacher functions. For any large integer $k$, ...
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2answers
82 views

Lebesgue measure-preserving differentiable function

Let $\lambda$ denote Lebesgue measure and let $f: [0,1] \rightarrow [0,1]$ be a differentiable function such that for every Lebesgue measurable set $A \subset [0,1]$ one has $\lambda(f^{-1}(A)) = ...
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1answer
31 views

Lebesgue measure of union of semi-open interval

Given $\mathbf{A} = \bigcup_{n\geq0}[n,n+ \frac{1}{2^n}[$ and the Lebesgue measure $\lambda$, find $\lambda(\mathbf{A})$. My solution: \begin{align} &\lambda\left(\bigcup_{n\geq0}[n, ...
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1answer
29 views

When Borel functions and Baire functions are equal?

Suppose $X$ is compact metric space. Let $A$ be the smallest set of complex functions containing all continuous functions such that: If $f_n \in A$ are uniformly bounded and $f_n \to f$ pointwise ...
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28 views

Integral of a function $f: \mathbb{N}\times\mathbb{N} \to\mathbb{R}$

Let $X = Y = \mathbb{N}$, $A = B = P(\mathbb{N})$, $\mu$ and $\nu$ counting measures on $(X, A)$ and $(Y, B)$. Define $f:X\times Y \to \mathbb{R}$ by $f(m,m) = 1$, $f(m+1,m) = -1$ and $f(m,n) = 0$ ...
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0answers
33 views

Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0 < P(B) < 1$

Let $(\Omega, \mathbb{F}, P)$ be a probability space. Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0$ $<$ $P(B)$ $<$ $1$. My ...
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1answer
22 views

Function of bounded variation and integration

Let f belong to $C[a,b]$. Show that there is a function g that is of bounded variation on [a,b] for which $\int_a^bfdg=||f||_{max}$ and TV(f)=1. This problem appears on page 162 of Royden's Real ...
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1answer
25 views

If $\{f_n\}$ is Cauchy in measure, then there is a measurable function $f$, such that $\{f_n\}$ converges in measure to $f$

The theorem is from Real Analysis (Carothers). Let $\{f_n\}$ be a sequence of real valued measurable functions, all defined on a common measurable domain $D$. If $\{f_n\}$ is Cauchy in measure, then ...
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0answers
262 views

We have sums, series and integrals. What's next?

We know how to sum or average a finite number of terms: sums. We know how to sum a countable infinite number ${\beth_0}$ of terms: series. We know how to sum ${\beth_1}$ terms: integrals. How to ...
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2answers
44 views

Markov Inequality proof (measure theory)

I am trying to prove Markov's Inequality in measure theory as: Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be a non-negative function which satisfies $g(x)>0$ se $x>0$, and not descendant in ...
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1answer
21 views

Characterization of support of positive regular Borel measures

Let $\mu$ be a positive Borel measure ona compact Hausdorff topological space. I am trying to prove the following: Show that $x \in support(\mu)$ if and only if $\int_X f d \mu >0$ for every ...
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20 views

$x \in supp(\mu)$ iff $\int f d\mu >0$ for every $f \in C_c(X,[0,1])$ with $f(x)>0$

I'm reviewing for a real analysis midterm and have a question about this problem ($\mu$ is a Radon measure). I have two separate solutions to the "if" part, but have a question about each one. ...
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1answer
16 views

Can the simple function in a product space be chosen this way?

It is a classical proof that if you have a positive function n any measure-space, there is a monotone sequence that converges pointwise to this function. If you have two measure-spaces $(\Omega, ...
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1answer
33 views

limit of powers of measurable positive function

This question appeared on a Measure Theory exam a couple days ago: Let $(X,\mathscr M,\mu)$ be a measure space. Suppose $\mathbf{f:X\to[0,\infty)}$ is positive and measurable. Define a measurable set ...
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63 views

Proof that $f = 0$ almost everywhere

My question is about the proof of parts (a) and (b) of Theorem 1.39 on page 30 of Rudin's "Real and Complex Analysis." 1.39 Theorem. DIFFICULTY # 1: (a) Suppose that $f : X \to [0, \infty]$ is ...
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1answer
47 views

Lebesgue measure of an intersection of a sequence of subsets

This is exercise 1.19 from "A User-Friendly Introduction to Lebesgue Measure and Integration" by Gail S. Nelson, and $m(E)$ is notation for Lebesgue measure of set $E$: Let ${E_{k}}$ be a sequence of ...
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0answers
20 views

Equivalent conditions for weak $L^p$ spaces for $p\leq 1$

I have difficulty doing the following exercise from Tao's real analysis book: Let $X$ be $\sigma$-finite measure space and $0<p\leq 1$. Then show that the following are equivalent: $f$ is in ...
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2answers
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Let $X, Y$ be topological spaces and let A ∈ B($X$) (Borel $\sigma$ algebra on $X$), B ∈ B($Y$). How to show that A × B ∈ B($X\times Y$)?

Let X be a topological space. All that I know is Borel $\sigma$ algebra on X is the smallest $\sigma$ algebra generated by $T_X$ i.e. set of all open sets in X. Is there any other characterization of ...
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0answers
55 views

Capacity vs measure of a set - intuitive understanding

There is a concept of measure of "largeness" of a set, called capacity. The intuition is, instead of physical largeness (measured by Hausdorff or Lebesgue measure), capacity measures how good a given ...
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29 views

Semicontinuity of stochastic kernel

Let $X$ and $Y$ be metric spaces with Borel sigma algebra and $P(B|y)$ be a stochastic kernel on $X$ given $Y$. I'm trying to proof the equivalence of the following two statements: (i) $P(\cdot,y)$ ...
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18 views

What are the measurable sets induced by this outer measure?

This is exercise 1.3.1 in Cohn, Measure Theory, second edition. I think I have solved it correctly, but would appreciate verification. The problem is mentioned in this thread, but it's part of a ...
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1answer
19 views

Convergence of Random Variable (Measure Theory)

If $W_n$ has a Poisson(n) distribution, how to show $X_n=(W_n-n)/\sqrt{n}$ goes to N(0, 1) by using some measure theories?
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2answers
35 views

$E_{k} \subset [0,1]$ such that $\lim_{k \to \infty} m(E_{k}) = 1$ but $\bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}E_{k} = \phi $

I am trying to find an example of collection $\left \{ E_{k} \right \}_{k=1}^{\infty}$ such that each $E_{k} \subset [0,1]$ satisfying $\lim_{k \to \infty} m(E_{k}) = 1$ but ...
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3answers
63 views

How to show that in a subset of [0,1] of measure greater than 0.5, there exist two points at distance exactly 0.1?

My attempt: Let's disregard isolated points as they do not contribute to measure. If our set is union (disjoint) of finitely many, say $n$ intervals, there must be at least ($n-1$) intervals of ...
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Convergence of multiple integrals [duplicate]

The question is as follows: Let $f:[0,1]\rightarrow R$ be continuous. Prove $lim_{n\to\infty}\int_{0}^{1}\int_{0}^{1}...\int_{0}^{1}\int_{0}^{1}f((x_1+x_2+...+x_n)/n)dx_1...dx_n=f(1/2)$ I'm ...
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1answer
51 views

Show that there exists a continuous function $f$ such that $\int |\chi_A-f| d\lambda\lt \epsilon$

Let $\lambda=l^*$ denote Lebesgue measure on $\Bbb R$, and let $A$ be a Lebesgue measurable set with $\lambda(A)\lt +\infty$. Show that if $\epsilon \gt0$, there exists an open set which is the union ...
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0answers
6 views

General set in product space approximated by rectangle sets

Let $(E^k,\mathcal{E}^k,\mu^k)$ be a product measure space. By a rectangle set in $E^k$, we mean a set of the form $A_1\times\ldots\times A_k$ where each $A_i\in \mathcal{E}$. My question is, for ...
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1answer
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Question about Vitali Covering (from a Lemma in Royden and Fitzpatrick's book)

Definition. For a real valued function $f$ and an interior point $x$ of its domain, the uppper derivative of $f$ at $x$ denoted by $\overline{D}f(x)$ is defined as follows: ...
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0answers
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help with a proof on Doob's Submartingale inequality - application of chebychev's inequality

I am stuck on a final step of the proof, we have that $(X_n)$ are non negative submartingale, and $c>0$. We let $T = \inf \{n: X_n > c \} \wedge N$ which is a stopping time. Let $E \{ ...
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0answers
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Comparing Patrick Billingsley's Aniversary Edition to previous editions, and to Robert B. Ash's book.

I'm reading some of the reviews at amazon to the Anniversary edition of Billingsley's 'Probability and Measure', and several users state that the book is riddled with new typos, and plain errors, ...
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0answers
19 views

Quotient of measurable functions

I need to show that if $f,g : X \to \mathbb{R}$ are measurables with respect to the $\sigma- algebra$ $S$ of $X$, and $g(x) \neq 0, \forall x \in X$, then $f/g: X \to \mathbb{R}$ is measurable. So ...
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0answers
23 views

Weak convergence in probability and functional analysis

Let $X$ be a metric space. By definition, the sequence of Borel measures $\mu_n$ on $X$ converges weakly to a measure $\mu$, if for all bounded continuous functions $f:X\to\mathbb{R}$ we have ...
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2answers
47 views

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function that is in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$. Here's what I have so far. $f\in L^2 ...
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23 views

Show $A$ is an algebra of sets of $X$ [duplicate]

I'm having trouble with this problem, and not sure where to start with. I'm not sure if this is related to Borel algebra since they have very similar construction. Could someone help me with the ...
0
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1answer
11 views

Functions h and r: prove $||h||_p = sup_{||r||_q \le 1} \int_E{rh}$

Functions h and r: prove $||h||_p = sup_{||r||_q \le 1} \int_E{rh}$, where $1/p + 1/q =1$. Using Holder's Inequality, I can prove that $||h||_p \le sup_{||r||_q \le 1} \int_E{rh}$, but I'm having ...