# Tagged Questions

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### Another version of the Poincaré Recurrence Theorem (Proof)

The task is to prove the following version of Poincaré's Recurrence Theorem: Let $(X,\Sigma,\mu)$ be a finite measure space, $f\colon X\to X$ a measurable transformation that preserves the ...
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### A question on Abstract measure spaces

Let $(X,M)$ be a measurable space then 1) if $\mu$ and $\lambda$ are measures in $M$ st $\mu \ge$ $\lambda$ then show that $m$ defined as $\mu= \lambda + m$ is a measure 2) Prove that if ...
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### Prove that a function with a measurable graph is differentiable [closed]

Let $f:[a,b]\to(0,\infty)$ be continuous and let $Gf:=\{(x,y)|y=f(x)\}$ be the graph of $f$. Prove that $Gf$ is measurable only if $f$ is differentiable in $(a,b)$?
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### When is $\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$

An elementary question on Riemann - Integration: Under what conditions on $f$ is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If $f$ is bounded in $[a,b]$, then this is ...
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### Find Jordan decomposition constructed by function.

$$F(x) =\left\{ \begin{array}{l l} 0 && x \leq 0\\ 1-x && 0 \leq x \leq 1\\ 2x-4 && 1 < x \leq 2 \\ 1 && 2 < x \end{array} \right.$$ ...
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### Problem regarding outer measure

Let $\mu^*$ be an outer measure on a set $X$. We want to show that $E\subset X$ is $\mu^*$-measurable iff for every $\epsilon>0$ there exists a $\mu^*$-measurable set $F\subset E$ such that ...
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### Showing a set function to be measure

Let $(X,\mathcal A)$ be a measurable space and let $\mu:\mathcal A\to [0,+\infty]$ be a finitely additive set function such that $\mu(\emptyset)=0$. We want to prove that $\mu$ is a measure on ...
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### Measure Theory question 4

If f is a non negative mesaurable function on $\mathbb R$ such that $\int f<\infty$, show that the set $\{x|f(x)>0\}$ can be written as a union of an ascending sequence of measurable sets of ...
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### Identify the smallest sigma-algebra of subsets of $\mathbb{R}$ that contains the set [0, 1]

This is a past exam question which I've tried to do this myself, though I'm unsure of the solution. First of all, by the definition of a sigma algebra, I should include $\emptyset$. Then, ...
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### Monotone convergence, measure-theory, is this excercise correct?

Here is the exercise: I have some questions: Is this correct when k starts with 1?, the Taylor series with e starts with 0? But does the zero disappear in some way?, I can not see how. I know that ...
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### Minkowski sum of a positive Lebesgue measure set and $\mathbb{Q}$.

Let $A\subset \mathbb{R}$ be of positive Lebesgue measure, i.e. $\mu(A)>0$. Is it then true that $\mu(\mathbb{R}\setminus (A+\mathbb{Q})) = 0$? I am quite sure that if $\mu(A)>0$, then $A-A$ ...
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### Lebesgue measure problem

Let $f$ be a non-negative measurable function on $\mathbb{R}$, and suppose that $\int f=0$. Prove that the set where $f \neq 0$ is a zero set. The hint says to let $E_n=\{f>1/n\}$ and then compare ...
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### Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to ...
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### f being a Lebesgue integrable function on $(0, a)$ implies that $g(x) = \int_x^a (f(t)/t)dt$ is also integrable.

I need to prove: If f is Lebesgue integrable on $(0, a)$ and $g(x) = \int_x^a (f(t)/t)dt$, then g is integrable on $(0, a)$. I know that since f is integrable on the interval $(0, a)$ I have ...
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### lebesgue measurable problem

Let $E$ be a Lebesgue measurable subset of $[0,1]$, and suppose that $m(E)>3/4$. Prove that $(-1/2,1/2) \subseteq E-E$. We use $E-E$ to denote the set $\{x-y:x,y \subseteq E\}$
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