# Tagged Questions

For questions about the Lebesgue measure, a measure defined on the Borel or Lebesgue subsets of the real line or $\mathbb R^d$ for some integer $d$. Use it with (tag: measure-theory) tag and (if necessary) with (tag:lebesgue-integral).

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### Piecewise-$C^k$ function

A function $f:\mathbb{R}_+\to \mathbb{R}^n$ is called piecewise-$\mathcal{C}^k$ if There is a partition $\{ x_i \}_{i=0}^\infty$ with $x_0=0$ and $x_{i+1}>x_i$ such that $f$ is $\mathcal{C}^k$ ...
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### Fubini's theorem for a measure on product space, which is not a product of measures

Let $X,Y$ be some nice measurable spaces (I'm interested in $[0,1]$ so we can assume compact, etc.). Let $\mu$ be a measure on $X\times Y$ (again, assume it's a nice probability measure, or even ...
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### Is a measure on product space necessarily a product of measures?

Let $X,Y$ be some nice measureable spaces (i'm interested in $[0,1]$ so we can assume compact, etc.). let $\mu$ be a measure on $X\times Y$.(again, assume it's nice, i.e. probability measure. anything ...
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### What is the cardinal of non measureable set?

we know that $|P( \Bbb{R} )|=|L (\Bbb{R} )|$ ( $L (\Bbb{R} )$ is the set of all Lebesgue measureable set ) note that $L (\Bbb{R} ) \subsetneq P( \Bbb{R} )$. What is the cardinal of non ...
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### Evaluating $\lim_{n\to\infty}\int_0^n(1-(x/n))^ne^{x/2}dx$

$$\mbox{How to compute}\quad \lim_{n \to \infty}\,\,\int_{0}^{n}\left(1 -{x \over n}\right)^{n} \,\mathrm{e}^{x/2}\,\,\mathrm{d}x\,\,\, ?.$$ No ideas how to start this one. I see that the limit of ...
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### $\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin y}{ny(1+n^2y^2)}ndy$ via DCT?

I'm looking to calculate these limits/integrals: $$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin (x/n)}{x(1+x^2)}dx$$ 2.$$\lim_{n\to\infty}\int_0^{\infty}\dfrac{\sin(x/n)}{(1+x/n)^n}dx$$ I posted ...
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### For which value of “d” cantor set has lebesgue measure zero?

I tried to generalise the cantor set. I retain "$d<1$" size interval at both end and removed "1-2d" size interval, and process is same further. I have curious about what value of "d" this cantor ...
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### What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?

Suppose $\epsilon \in (0, 1)$ and $m$ is Lebesgue measure. What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?
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### Lebesgue measure, do we have $m(x + A) = m(A)$, $m(cA) = |c|m(A)$? [on hold]

Suppose $m$ is Lebesgue measure. Define $x + A = \{x + y : y \in A\}$ and $cA = \{cy : y \in A\}$ for $x \in \mathbb{R}$ and $c$ a real number. Let $A$ be a Lebesgue measurable set. I have two ...
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Let $m$ be Lebesgue measure. What is an example of Lebesgue measurable subsets $A_1, A_2, \ldots$ of $[0, 1]$ such that $m(A_n) > 0$ for each $n$, $m(A_n \Delta A_m) > 0$ if $n \neq m$, and $m(... 1answer 37 views ### What does a random variable 1 with subscript [0,1/2] mean? I came across the following notation that I cannot follow:$1_{[0,1/2]}$It is supposed to be some kind of random variable (or just an event? not sure) It is hard to google this, too. What does such ... 1answer 34 views ### Meaning of measurable function [closed] I don't understand the meaning of Measurable function , my lecture told us that$f(x)$measurable on measurable set$E$if$E(f>A)=\{x\in X:f(x)>A\}1.\quad$Can you please give me examples ... 1answer 54 views ### Do we have$\|x\|_p \le \|x\|_q$, where$x$is bounded and$p>q$? [closed] Assume that$x\in L_p(0,\infty) \cap L_q(0,\infty)$, where$p>q$. Do we have $$\|x\|_p \le \|x\|_q?$$ where$x$is bounded. 2answers 56 views ### What is the infimum/supremum of a set in the extended real line? May be a simple question, but I still don't know what the definition of the supremum or infimum of a set in the extended real line is. For example when we define the Lebesgue measure, we define the ... 1answer 20 views ### Inferring absolute continuity of summands from absolute continuity of sum Suppose we have i.i.d. random variables$X_1, X_2$. If$\text{Law}(X_1 + X_2)$is absolutely continuous with respect to the Lebesgue measure$\lambda$, can we infer that each$X_i$is absolutely ... 0answers 33 views ### Hölder inequality application to show that f=1 I want to proof that if$f \in L^{1}_{\mu}(\mathbb{R}), f > 0$continuous, satisfies$(\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{...
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Let $\mu$ and $\nu$ be two finite Borel measures on $\mathbb{R}$. We know that if $$\int f d\mu = \int f d\nu$$ for all continuous functions $f$ then $\mu=\nu$ and so the equation above holds for ...
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### Lebesgue outer measure is countably subadditive but not finitely additive proof

I have read all the Qs on this but couldn't find a clear proof. How can I prove that Lebesgue's outer measure is not finitely additive? Thanks! Edit: I understand I must show that the measure of the ...