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

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Polar decomposition of a measure

Prove : If $\nu$ is a complex-valued measure on a measurable space $(X,S)$, then there exists a non-negative real-valued measure $\mu$ on $S$, and a complex-valued measurable function $u$ on $X$ such ...
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1answer
17 views

Countable partition of a probability space

I am trying to prove the following statement. Even though it seems almost obvious that it must be true, I am having trouble with making my arguments precise. Let $\{D_i: i\in \mathbb{N}\}$ be a ...
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21 views

Trouble to calculate a $\| \;\; \|_{L^1}$ norm ($\int\limits_0^1 n e^{-nx}dx$)

Let $I = (0, 1)$ and $f_n(x) = n e^{-nx}$ a sequence of functions. I calculated $\|f_n\|_1$ and I obtained two different aswers. $$\|f_n\|_1 = \int\limits_0^1 |f_n(x)| dx = \int\limits_0^1 n e^{-nx} ...
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1answer
15 views

measurable function and properties of their integrals [closed]

I have to prove that given $(X,M,\mu)$ a measurable space and two measurable functions $\omega_i\colon M \rightarrow [0,\infty]$, $i=1$, $2$, if for all measurable sets E holds $\int_{E} \omega_1 = ...
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1answer
25 views

Measure of open sets covering compact set

Prove that if $F$ is a finite collection of open intervals that covers a compact interval $[a, b]$, then the sum of the lengths of the intervals in the collection is strictly greater than $b − a$ ...
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1answer
31 views

Components of $\sigma$ algebras

I previously thought that $\mathcal{B}(\mathbb{R})^n = \mathcal{B}(\mathbb{R}^n)$, which I just realized was false! I'm wondering whether the following weaker statements are true: Let $\mathcal{F}$ ...
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2answers
42 views

Rudin's proof of Riesz representation theorem

Let $X$ be a locally compact Hausdorff space. Let $C_c(X)$ be the set of complex valued continuous functions of compact suppot on $X$, $C_c^+(X)$ the set of non-negative real valued functions in ...
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39 views

Sum of not necessarily independent discrete random variables.

Let ${X_k}$ be a sequence of discrete random variables, where $P(X_k=k)=\frac{1}{k^2}$ and $P(X_k=0)=1-\frac{1}{k^2}$. Let $S_n=\sum_{i=1}^n X_i$. Does $\frac{S_n}{\sqrt{\log n}}\rightarrow 0$ (in any ...
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1answer
59 views

How to compare the Hardy-Littlewood maximal function for balls and cubes?

I am currently working through a set of notes I found on the internet at: http://math.msu.edu/~charlesb/Notes/DuoChapter2.pdf I am up to page 8, and the Hardy-Littlewood maximal function for balls ...
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31 views

Random variables and the topology of weak convergence

To see what's going on, I am trying to translate the idea of topology of weak convergence on a random variable setting, just to get some concrete intuition. This is what I have got so far (where the ...
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1answer
28 views

Is there a measure invariant with respect to the Möbius transformation?

I would like to use a measure ${\rm d} \mu (z)$ on ${\mathbb C}$ so that for any $f(z)$ $$\int_{\mathbb C} f(z) {\rm d} \mu (z)$$ is invariant under Möbius transformations. Taking the ...
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23 views

What does it mean “rotationally invariant density”?

In the great answer given by the math.SE user @Tim, he does 2 hypothesis, on of the which ones is about the rotationally invariance of the density. Can you explain formally what does it mean? I do ...
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2answers
18 views

Mean value theorem for sliding window of Lebesgue integral of integrable function

Take $f \in L^1(\mathbb{R})$ and define $g(x) = \int_x^{x+1} f(t) \, dt$. If $g(a) > 0$ and $g(b) < 0$, is it necessarily true that there is some $c \in [a,b]$ such that $g(c) = 0$? I feel as ...
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2answers
41 views

$f(x-y)$ considered as a function of $(x,y)\in \mathbb{R^{2n}}$ is measurable if $f$ is measurable

I know there are similar questions up proving this, but I had a question specific to the following proof (specifically in bold): Let $f$ be a Lebesgue measurable function on $\mathbb{R^n}$. Then the ...
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2answers
80 views

If a function is Lebesgue measureable, does this imply Lebesgue integrability?

Say we take the measure of a countable set, we obtain that $\mu=0$. Now if this is the case, does this automatically imply that it is Lebesgue integrable as well? The reason I bring up the set is ...
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3answers
99 views

a interesting question from topological group

$G$ is a locally compact Hausdorff topological group, $m$ is a (left) Haar measure on $X$, $A$ and $B$ are two finite positive measure in $G$, that is $m(A)>0$, $m(B)>0$. My question is: Can ...
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1answer
38 views

Show that there exists a measurable function $g$ with $|g| \leq 1$ so that $\int_E g d\nu = |\nu|(E)$ for all measurable sets $E$.

Let $(\mathcal{X},\mathcal{M},\nu)$ a measure space where $\nu$ a signed measure. Show that there exists a measurable function $g$ with $|g| \leq 1$ so that $\int_E g d\nu = |\nu|(E)$ for all ...
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1answer
41 views

On the measurability of a special set

Is $\underset{n=1}{\overset{\infty}{\cup}}[-n,n]$ measurable? This is a question on Measure Theory of a previous exam period. I can't understand if it's that easy as I find it, or I'm missing ...
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1answer
23 views

Coniditional expectation for bounded random variables

I need to show that for bounded random variables $X$ and $Y$ we have ${\rm{E}}[X{\rm{E}}(Y||\mathcal{F})]={\rm{E}}[Y{\rm{E}}(X||\mathcal{F})]$. The only property of conditional expectation I am aware ...
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2answers
37 views

F measurable and conditional expectation.

(a):I found it easily cause sum of measurable sets are measurable. (b),(c): I know limsup(Sn/n) is also measurable but I can't prove that just sup(Sn/n) is measurable. (d): I solved it by using the ...
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1answer
14 views

Complete separable metric space X represented represented as union of closed sets

I have a problem concerning a statement I found in volume 2 of the classic reference book on measure theory by Bogachev. More precisely, I have a problem concerning theorem 6.1.13. I the proof the ...
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1answer
38 views

Measure space $(X,\mathcal{F},\mu)$ where $L^p(X,\mathcal{F},\mu) \neq L^q(X,\mathcal{F},\mu)$ if $p\neq q$

I was trying to solve this problem: Let $(X,\mathcal{F},\mu)$ be a measure space where $L^p(X,\mathcal{F},\mu) \neq L^q(X,\mathcal{F},\mu)$ when $p\neq q$. Prove that there exist a sequence of sets ...
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25 views

Density of simple functions in $L^\infty$

$\textbf{Theorem}$ Is the set of simple measurable functions $s$ (with values in $\mathbb{R}$ or $\mathbb{C}$) such that $ \mu(\{x: s(x)\neq 0\}) <\infty$ is dense in $ ...
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1answer
26 views

Why do we use non-negative functions in measure theory

I've just started learning Measure theory and I was curious if there is a rationale for working with non-negative functions as the 'base' upon which more complex theorems are built. Why not include ...
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22 views

Lebesgue integral over subset

I have a basic question related to Lebesgue integration restricted to a subset of $\mathbb R^n$. In general, for $(X,\Sigma,\mu)$ a measurable space, if $f$ is measurable one can calculate the ...
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1answer
32 views

Measurable set as a subset of a non-measurable set

is it possible to find a measurable subset $A$ of a non-measurable set $B$ such that $A$ can be assigned the measure zero or more particularly a non zero value ( in the case of the Lebesgue measure ...
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0answers
21 views

A property on exterior measures on a metric space

Let $(X,d)$ be a metric space and $\mu:2^X\to[0,\infty]$ be an exterior measure on $X$, $\mathcal{M}:=\{E\subset X:\forall S\subset X\ \ \ \mu(S)=\mu(S\cap E)+\mu(S-E)\}$. Suppose ...
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26 views

Kolmogrov 0-1 law

Let {$A_n$} be an independent sequence of events. Show that the event $$(\omega:\frac{\sum_{k=1}^n I_{A_k}(\omega)}{n} \rightarrow x)$$ has probability either 0 or 1. I think if I show this event as ...
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2answers
47 views

Non-measurable subsets of a null set

Given the Lebesgue measure on $\mathbb{R}$, I am wondering if a non-measurable set can always be included in a null set? More precisely, let $A$ be a set in the Borel σ-algebra ...
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37 views

Inner regularity of the Haar measure

Let $G$ be a locally compact Hausdorff topological group, and denote by $B$ the σ-algebra generated by the open subsets of $G$, an element of $B$ is called a Borel set. A (left) Haar measure m ...
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1answer
20 views

Kolmogorov 0-1 law, Measure Theory

Suppose that $(X_1, X_2,...)$ is an independent sequence of random variables and $Y$ is measurable $\sigma(X_n,X_{n+1},....)$ for each $n$. Show that there is a constant a such that $P(Y = a) = 1$. I ...
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2answers
31 views

Difference between open sets and open balls in metric space

Let $X$ be a separable metric space and let $\mathfrak{M}$ be the $\sigma$-algebra generated by open balls in $X$. Show that $\mathfrak{M}$ contains all the open sets in $X$ and all the closed ...
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2answers
41 views

Application of Monotone Convergence Theorem

Suppose $f ∈ L^{1}([0, 1])$. Prove that $lim$ $ε→0^{+} \int_{[0,ε]} f dµ = 0$ My attempt at proof: Let $B_N$ be an open ball of radius $N$ centred at origin. $E_N:=$ {$x: f(x)\leq N$} ...
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Integral of a function over the Koch Curve. Is it rigourous enough?

(I want to investigate the validity of this approach, as I already know this is the correct result) I present a proof that $$\int_{K} (x+y) \ \mu(x,y)={{9+\sqrt 3} \over 18}$$ Where the region of ...
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1answer
30 views

Does it hold $\sigma(X_0,\ldots,X_n)=\sigma(X_1,X_1-X_0,\ldots,X_n-X_{n-1})$?

Let $(\Omega,\mathcal{A})$ and $(\Omega',\mathcal{A}')$ be measurable spaces $X_1,\ldots,X_n$ be measurable with respect to $\mathcal{A}$-$\mathcal{A}'$ $Y_m:=X_m-X_{m-1}$ for $1\le m\le n$ and ...
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1answer
30 views

The geometry meaning of Riemann–Stieltjes integral [duplicate]

Maybe my question seems so strange but I want to know what is the geometry meaning of Riemann stieltjes integral ??
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2answers
42 views

Isn't the statement of the Fatou's lemma somewhat problematic?

My lecture notes define $\int f := \int f^+ - \int f^-$ provided both $\int f^{\pm}$ are finite. And then the Fatou's lemma is stated in the following way: Let $f_n$ be a sequence of integrable ...
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1answer
31 views

Almost everywhere differentiability

Suppose $f: \mathbb{R} \to \mathbb{R}$ is increasing and $g = f$ almost everywhere with respect to Lebesgue measure (a.e.). Suppose $g'$ exists a.e. Does it follow that $g' = f'$ a.e.? This comes ...
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1answer
24 views

If $X$ has density, when has $X\cdot I_A$ a density?

Let $(\Omega, \mathcal F, P)$ be a probability space, and $X$ be a random variable with some density function $f_X$. If $A \in \mathcal F$, then the indicator function $I_A$ has, as a discrete random ...
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1answer
21 views

Does it hold $\sigma(X_1,\ldots,X_n)=\sigma(X_n-X_1,\ldots,X_n-X_n)$?

Let $(\Omega,\mathcal{A})$ and $(\Omega',\mathcal{A}')$ be measurable spaces $X_1,\ldots,X_n$ be measurable with respect to $\mathcal{A}$-$\mathcal{A}'$ $Y_m:=X_n-X_m$ for $1\le m\le n$ I'm ...
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336 views

What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve

What is $$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$ where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure look here. Attempt: I can evaluate the integral numerically and I have derived ...
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1answer
30 views

Prove that the image of a curve has zero content

Definition: A set $A \subset \mathbb{R}^2$ is said to have zero content if, for all given $\varepsilon >0$, exists a finite collection of rectangles $A_1, \dots, A_n$ such that $A \subset ...
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2answers
61 views

The norm $\|f_n-f\|_{L^1} \to 0$ but $f_n \not\to f$

A classmate and I are studying this following question from Stein-Shakarchi, Chapter 2, Exercise 12: Show that there are $f \in L^1(\mathbb{R}^d)$ and a sequence $\{f_n\}$ with $f_n \in ...
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1answer
38 views

If E is measurable, then $\delta E$ is measurable.

Problem: If $\delta =(\delta_1,\delta_2,\cdots,\delta_d)$ is a d-tuple of positive numbers $\delta_i>0$, and $E$ is a subset of $\mathbf{R^d}$, we define $\delta E$ by $\delta E = ...
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Characterisation of continuous functions whose support is compact

I saw this question some time ago but I can't remember where. Basically the question ask to find the set of all continuous functions $f:\mathbb{K} \to \mathbb{R}$ whose support $\text{supp}(f) := ...
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1answer
26 views

Why are these two definitions of Lebesgue outer measure equivalent?

I got two definitions of Lebesgue outer measure one is $$\mu^*(A)=\inf\Big\{\sum^{\infty}_{k=1}l(I_k):A\subset \bigcup_k I_k\Big\}$$ and the other is $$\mu^*(A)=\inf\Big\{\sum^{\infty}_{j=1} ...
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3answers
45 views

Prove if $E_1$ and $E_2$ are measurable then $m(E_1\cup E_2)+m(E_2\cap E_2)=m(E_1)+m(E_2)$

by additivity $m(E_1\cup E_2)=m(E_1)+m(E_2)$ (because $E_1,E_2$ are measurable) but i don't know what to do with $E_1\cap E_2$. I tried to use demorgan's identity to solve this part but this is not ...
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1answer
43 views

Problem on measure theory

Let $ \mu $ be a $ \sigma $-finite measure space on $(X,s)$. Suppose $ f: X \to [0,\infty]$ be a $ s $-measurable and $ p \in [0,\infty]$. Show that $$ \int_X f^p \, d\mu = \int_{0}^{\infty} pt^{p-1} ...
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1answer
15 views

Application $\pi$-$\lambda$ lemma one-sided Markov shift

Let $(S_k^{\mathbb{N}},\Sigma_k^{\mathbb{N}},m,\tau)$ be the probability preserving transformation of the one-sided Markov shift, where $\Sigma_k^{\mathbb{N}}$ is the $\sigma$-algebra generated by the ...
2
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0answers
19 views

pythagorean theorem for conditional experience

Let G be a subsigma algebra and X is squareintegrable: => $ E[X²] = E[(X-E[X|G])²] + E[E[X|G]²]$ I know that this can be directly shown interpreting the conditional experience as a projection in ...