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

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Bounding below Hausdorff measure of conected set

I'm trying to prove that for every connected set $E\subset\mathbb{R}$, $H^1(E)$, the Hausdorff measure is bounded below by $\text{diam}(E)$. In the answer there, it was suggested to use a Lipschitz ...
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1answer
11 views

Can we bound a Brenier map between uniform distributions with the Hausdorff distance between their supports?

Let $A,B$ be compact subsets of $\mathbb{R}^n$. Let $\mu_A$ (resp. $\mu_B$) be uniform probability measures over $A$ (resp. $B$). Then as a consequence of Brenier's theorem there is a one-to-one ...
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0answers
14 views

Volume Zero of Not Continuous Function

Show that a bounded real-valued function f on a closed interval $I$ of $E^n$ is integrable on $I$ if and only if the set of points of $I$ at which $f$ is not continuous is the union of a sequence of ...
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2answers
25 views

Proof outer measure satisfies monotonicity: $A \subseteq B \implies m^*(A) \leq m^*(B)$

Theorem: $$A \subseteq B \implies m^*(A) \leq m^*(B)$$ Proof Attempt: By definition, $m^*(B) = \inf\{\sum\limits_{k=1}^\infty |J_k||\{J_k\} \text{ is a cover of B }\}$, $m^*(A) = ...
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24 views

Given two finite measures $\mu$ and $\lambda$ on $R$, prove the following:

Given two finite measures µ and λ on R, show that (a) $\nu = λ + µ$ is a finite measure (b) $µ << \nu$ and $λ << \nu$ (c) Write $$\mu(E) = \int_E f d\nu$$ $$λ(E) = \int_E g d\nu$$ Let ...
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0answers
16 views

Question on bounded function and Radon Nikodym Derivative.

Given $\lambda << \mu$ finite measures, show that for any bounded function $g$ and $A \subset [0,1]$ $$\int_A gd\lambda = \int_A ghd\mu$$ where $h$ is the Radon-Nikodyn derivative ...
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0answers
21 views

Finding conditions on functions for absolute continuity of measure

Let $\mu$ and $\lambda$ be finite measures of $[0,1]$ which are absolutely continuous with respect to Lebesgue measure: $\lambda << m$ and $\mu << m$ $$\lambda(A) = \int_Af(x)dx$$ ...
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1answer
51 views

Does length $[0,1]$ = length $(0,1)$?

So we know that the length of an interval $[a,b]$ is simply $b-a$ but does this hold if the interval is open? Or if one of the sides are open, like $(a,b]$ or $[a,b)$? Also, can I just confirm that ...
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1answer
21 views

Continuous function differentiable almost everywhere, show f' is measurable

$f:\mathbb{R} \to \mathbb{R}$ continuous and differentiable almost everywhere, show that $f'$ is measurable. I feel like I should use some sequences, but I don't know how to start. Anyone?
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1answer
24 views

Monotonic increasing and convergence in measure

If for each $n\in\mathbb{N}$, $f_n$ is monotonic increasing on [0,1] and $f_n\rightarrow f$ in measure, then $f_n\rightarrow f$ at every x at which f is continuous. I'm not sure whether this is right ...
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0answers
15 views

Relationship between Convergence in mean, convergence in measure and a.e. convergence

What is the relationship between convergence in mean under 1-norm (http://mathworld.wolfram.com/ConvergenceinMean.html), convergence in measure and a.e. convergence? I have shown that convergence in ...
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1answer
41 views

Using the limit definition to Prove that a Set has a Vitali Covering

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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1answer
28 views

convergence of a subsequence of sequence converges in measure

If a sequence $(f_n)$ converge in measure to a function $f$, then every subsequence of $(f_n)$ converge in measure to $f$. Let $g_{n_k}$ a subsequence of $(f_n)$ then $|g_{n_k}(x)-f(x)|\leq ...
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0answers
19 views

Convergence in $L_p$

Let $f_n=n\chi_{[1/n,2/n]}$ in $\mathbb{R}$ with the Lebesgue measure defined on the Borel subsets of $\mathbb{R}$. Show that $f_n$ does not converge in $L_p$ to the $0$-function. $\left(\int (f_n)^p ...
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1answer
26 views

Measure of a set, convergence in measure

Let $(f_n)$ be a sequence of measurable real-valued functions which is Cauchy. Select a subsequence $(g_k)$ of $(f_n)$ such that the set $E_k=\{x\in X: |g_{k+1}-g_k(x)|\geq 2^{-k} \}$ is such that ...
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1answer
21 views

taking a limit inside a measure

dealing with the probability measure, I have that $$P(\cap_{n=1}^\infty A_n) = \lim_{M\to \infty}P(\cap_{n=1}^M A_n)$$ for some evens $A_n$. Could someone explain why we can do this?
2
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0answers
14 views

Linear transform of a strictly stationary time series

First, let me clarify what I mean by a strictly stationary time series. Let $(X_t)_{t\in \mathbb{Z}}$ be a sequence of random variables on some probability space. If it holds that $$(X_t, ...
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1answer
27 views

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
23 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 ...
2
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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 ...
2
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1answer
54 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 ...
2
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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
93 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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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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0answers
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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264 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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0answers
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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0answers
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
21 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
15 views

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 ...