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

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Measurable Subsets + Caratheodory Measurability

1.) What can go wrong if one assigns a measure to more subsets, especially to all subsets? (I would like to understand the subtleties behind) I imagine the first problem is to give the new subset ...
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1answer
85 views

References for a second course in probability theory

I need a probability book that treats all the arguments from the point of view of the measure theory and the Lebesgue integral. I've the basis of "naive" probability theory and of measure theory so I ...
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69 views

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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239 views

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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54 views

In Egorov's theorem, remove the condition $\mu(E) < \infty$ and let the sequence be convergent in measure. The conclusion holds for subsequence

Let $(X,\mathscr{F},\mu)$ be a measure space, $E \in \mathscr{F}$, $\{f_n\}$ is a sequence of measurable functions on $E$, and the sequence converges to function $f$ in measure. Show that $\exists ...
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397 views

Continuous, strictly increasing function that maps a set of positive (lebesgue) measure onto a set of measure zero?

Is there a continuous, strictly increasing (real-valued) function on the interval $[0,1]$ that maps some set of positive (lebesgue) measure onto a set of measure zero? Should I play with cantor ...
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96 views

Which are the conditions for a Lorentz space $L^{p,q}$ to be o-c?

Which are the conditions for a Lorentz space $L^{p,q}$ to be ord. continuous? ( A Banach function space is o-c $\equiv$ Increasing sequences of order-bounded positive functions converge in norm). ...
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52 views

Two measures on a same space

I have two measure space $(X, S, \mu_1)$ and $(X, S,\mu_2)$, where $S$ is the minimal $\sigma$-algebra containing sets $T = \{E_i\}_{i \in I}$. Suppose further that $T$ is closed under taking finite ...
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203 views

Must every probability distribution over a countable set be discrete?

Intuitively I expect this to follow from countable additivity, but there are ideas I can't rule out such as: Select a real number r from the uniform distribution over [0, 1]. If r is exactly 0.5, ...
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134 views

Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$ g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
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71 views

Entire function with $\int_{\mathbb C} |\log|f| |<\infty$

That's an old qual problem: If $\log|f|$ is absolutely integrable (with respect to $\mathbb R^2$ Lebesgue measure) with $f$ entire, then prove that $f$ is constant. One can see that ...
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78 views

A Lebesgue measurable universal Borel function

In 1918 Sierpiński constructed a Lebesgue measurable real-valued function on $[0,1]$ which isn't bounded above by any Borel function (I couldn't find the original reference, but here is a pdf of a ...
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1answer
52 views

Symmetrisation of function

Consider the probability space $\Omega = \{-1, 0, 1\}$ with the $\sigma$-algebra of all possible events and a probability measure $P$. Consider also the smaller $\sigma$-algebra $$F = \{\emptyset, ...
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47 views

Probability distribution “similar” to Gaussian.

Does there exist a distribution A other than Gaussian such that: 1) linear combination of random variable from A is distribution A 2) easy to integrate, for example find entropy Thank you
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44 views

Please explain this conditional expectation equality

I understand that E[X|Y] is a random variable. But I am kind of confused about the following : $$ \int_{\{Y=y_i\}} E[X|Y] dP = E[X|Y=y_i]P(Y=y_i) $$ In the above, P is a probability measure , then ...
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27 views

Why is this set an event?

As a part of a setup for another problem, my text remarks that it can be used without a proof that if $X_1, X_2, \ldots$ are random variables then $$C:=\{ \omega\in\Omega \ | \ \sum X_n(\omega) \ ...
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53 views

Application of DCT to show Mill's ratio for $N(0,1)$

Good day everyone, We want to show $\int_x^\infty e^{-t^2/2}dt \sim\frac{e^{-x^2/2}}{x}$ as $x\rightarrow\infty$ using the Lebesgue Dominated Convergence Theorem(DCT) for standard normal ...
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71 views

Why is this measure not finite, but $\sigma$-finite?

Let $\nu$ be the Lebesgue-measure on $[0,1]$, i.e. $\nu=\lambda_{|[0,1]}$ and $\mu(x)=\frac{1}{x}\nu(x)$. Show that $\mu$ isn't finite, but $\sigma$-finite. (1) In order to show, it isn't ...
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365 views

Weak convergence implies uniform convergence of characteristic functions on bounded sets.

Let $\{\mu_n:n\in\mathbb N\}$ and $\mu$ be distributions on $\mathbb R$, and let $\{\phi_n:n\in\mathbb N\}$ and $\phi$ be their respective characteristic functions. We can easily show using a direct ...
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64 views

$P[X_n\:\mathrm{converges}] = 0$ for iid non-constant RVs

I came across this problem while studying for an exam. If the sequence $\{X_n\}$ is iid Random variables and not constant with probability $1$, then $P[X_n\:\mathrm{converges}] = 0$. Apparently, ...
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160 views

A dense subalgebra of $C(X)$ that separates points

Any idea how to do this problem: If $X$ is a compact Hausdorff space and $A$ a subalgebra of $C(X)$ , where $C(X)$ is the algebra of all continuous functions, such that $A$ contains the constant ...
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86 views

Proof of outer measure

I have a kind of tricky question here that I would like to discuss with you. Define $\mu_0 : \mathbb{I} \to \overline{\mathbb{R}}_{+}$ by $\mu_0((a,b]) = F(b)-F(a)$ where $F$ is a weakly increasing ...
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44 views

$\sigma$-algebra defined by a function

It is easy to see that if $f:X\to Y$ is a function and $S_X$ is a $\sigma$-algebra over $X$ then $S_Y:=\{A\subseteq Y:f^{-1}(A)\in S_X\}$ is a $\sigma$-algebra over $Y$. Suppose now that $S_X=\sigma ...
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48 views

Is there any good text introducing a part of Borel-hierarchy which is in need in measure theory

Is there any good text introducing a subpart of Borel-hierarchy which is in need in measure theory, which can be done in short time? Say, 1~3 days if possible. (Assuming I'm studying about 14hours a ...
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84 views

Lebesgue measure on $I=[0,1]$

Can you help me with this :) $m$ is a Lebesgue measure on $I=[0,1]$, $g\in{L^{1}}(m)$ and $\int_{I} gf\, \mathrm{d}m=0$ for all $f\in{C(I)}$. Then I need to prove $g=0$ in $L^1(m)$?
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135 views

Is the following intersection of a set and a $\sigma$-algebra also a $\sigma$-algebra

Consider the statistical experiment of a dice roll. A probability space with $\left\{\Omega, \mathscr{F}, P\right\}$. Now $\Omega =\{1,2,3,4,5,6\}$, and $\mathscr{F}$ is the $\sigma$-algebra ...
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62 views

Is the function that gives you the measure of the neighborhood Borel?

Let $X$ be a compact metric space (with $\epsilon -$balls $B_{\epsilon }$) and $\mu $ a Borel probability measure. Let $a,\epsilon >0.$ Is the set $\left\{ x\in X:\mu (B_{\epsilon }(x))\geq ...
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41 views

Borel measures on $\mathbb{R}$ questions

I am reading a textbook and need some help. First it mentions that we can find a Borel measure such that $\int_\mathbb{R} x^2 \mu(x)<\infty$ but $\int_\mathbb{R} x \mu(x)=\infty$. This seems ...
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47 views

$C^\omega(\Omega)\cap C^\infty_0(\Omega)$.

Let $\Omega$ denote an open connected set in $\mathbf{R}$ (AKA open interval). Is it true and how can we prove it that $C^\omega(\Omega)\cap C^\infty_0(\Omega)$ consists of the zero function alone, ...
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76 views

$f\colon [a,b]\to\mathbb{R}$ Borel-measurable function: What does that mean?

A basic question: If I have a Borel-measurable function $f\colon [a,b]\to\mathbb{R}$, what is meant with that? Does this mean $\mathcal{B}(\mathbb{R})_{|[a,b]}-\mathcal{B}(\mathbb{R})$-measurable, ...
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1answer
27 views

An unknown function

I run into a function: $1_{[-n, n]^r}$. I guess this function equals 1 whenever x falls into $[-n, n]^r$. Am I right? I met this function in an analysis paper which deals with measure and density of ...
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49 views

Show that exists a not decreasing function that $f:(a,b)\rightarrow \mathbb{R}$ that is continuous only in $(a,b)\setminus D$.

Show that there exists a not decreasing function $f:(a,b)\rightarrow\mathbb{R}$ continuous on $(a,b)\setminus D$ and discontinuous on $D$ where $D$ is a countably infinite subset of $(a,b)$. This is ...
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84 views

Recommend me a text or webpage introducting gamma function throughly

Till now, i have learned abstract Integration, all basic properties of the (n-dimensional) Lebesgue(-Stieltjes) measure and the lebesgue integral is an extension of Riemann integral. Here's an ...
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83 views

When does a measurable function exist with a given distribution?

Let's suppose (A,X,P) and (B,Y,Q) are two probability spaces (A,B underlying spaces, X,Y sigma-algebras, P,Q probability measures, respectively). Under what (topological and/or measure theoretic) ...
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38 views

Is $\int g\, d\mathbb{P}=\int g\, d\mathbb{P}_{|\mathfrak{F}}$?

Let $(\Omega,\mathfrak{A},\mathbb{P})$ be a probability space and $\mathfrak{F}\subset\mathfrak{A}$ a sub-$\sigma$-algebra. Consider $g\in L_{\mathbb{P}}^1$ and $g$ $\mathcal{F}$-measurable. ...
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52 views

Show: The integral over a zero set is zero

From Show: $\mathbb{E}(f|\mathcal{F})=\mathbb{E}(f)$ was motivated: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space. Let $A\in\mathfrak{A}$ with $\mu(A)=0$. I would like to show that $\int_A f\, ...
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understanding simple functions

Let $(X,\mathcal{M})$ be a measurable space. The definition of a simple function on a set $X$ is that it is a finite linear combination, with real coefficients, of characteristic functions of sets in ...
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75 views

Existence of $1/i$-dense subsets

Let $(X, d)$ be a compact metric space and $m$ be a Borel measure on $X$. Assume that $\lbrace A_i\rbrace$ is a nested sequence of subsets: $\dots\subset A_i\subset\dots\subset A_2\subset A_1$ and ...
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52 views

Convergence in $L^p$ of $f_x(y)=f(y-x)$

Looking through practice problems to get ready for my exam, I found the following one which confuses me a bit: Let $1\le p<\infty$ and $f\in L^p(\mathbb{R})$. For $x\in\mathbb{R}$ let ...
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25 views

Is a locally constant function on the complement of a null-set measurable?

If I have a surface $M$ and a function $\ f: M \rightarrow \mathbb{R}$, which is locally constant on the complement of a measure-zero set, i.e., there is a measure-zero set $V \subset M$, such that ...
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56 views

aplication of convergence theorem for integrals

I am studying the proof of a theorem, and if my affirmation is true then I understand the proof of the theorem. I dont know how to prove the affirmation... Affirmation: Let $n \in N , n \geq 1$ and ...
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56 views

measurable subset of possibly non-measurable set.

Let $X$ be a Polish space. Let $T$ be a possibly uncountable set of Borel probability measures on $X$. Suppose there exists a subset $A$ of $X$ with the following property: For each $\mu\in T$, there ...
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83 views

Lebesgue measure of an image of a continuous function

There is given a function $f\colon\mathbb R\to\mathbb R$ that is $C^1$ and set $K$ so that $K = \{\,x\mid f'(x)=0\,\}$. How to prove that image of set $K$ is of measure $0$ (one-dimension Lebesgue ...
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54 views

Approximation property confusion

I am a bit confused about this: If $A$ is an algebra of sets then for any $B \in \sigma(A)$ there exists a sequence of $B_n \in A$ such that $P(B \Delta B_n) \to 0$ $A \Delta B $ is denoting ...
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162 views

$L^p$ spaces and counting measure

currently I am working on the following two exercises as a revision for my exam. Let $\mu$ be the counting measure on $\mathbb N$. Show that if $1 \le p < s < \infty$ then $f \in L^p$ implies ...
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114 views

Problem related to Fatou's Lemma (Measure theory)

Statement of the problem: Show that if $\{f_n\}_{n=1}^{\infty}$ is a sequence of non-negative measurable functions on $\mathbb{R}^d$, then for any $t>0$ we have: $$m\left(\left\{x\in ...
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75 views

Stopping time and filtrations

I have a definition problem. I know that a filtration on a probability space is an increasing sequence of $\sigma$-algebras. I was now thinking on the fact that constant times are stopping times. I've ...
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49 views

One implication (on Measure)

Please be noted that charges are finitely additive measures and measure are countably additive ones. Theorem 2.1. Let $\mu$ be a charge on a Boolean algebra $B$. Each of the following conditions ...
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89 views

Help proving that a measure is absolutely continuous with respect with respect to another measure

Suppose that $E$ is a locally compact and separable metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In ...
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46 views

Not convinced by condition for converg. in prob. $\implies$ converg. in mean.

I've found an exam question which gives a sufficient condition for the implication $X_n \overset{P}{\rightarrow}X \implies X_n \overset{L_1}{\rightarrow}X$, but I have a counter example which seems to ...