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

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3
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1answer
715 views

limit of increasing sequence of measures is a measure

Statement Let $(X,\Sigma,\mu)$ be a measurable space and let $(\mu_n)_{n\geq 1}$ be a sequence of measure in this space. Suppose that this is a monotone increasing sequence, in the sense that ...
4
votes
0answers
1k views

Meaning of “almost everywhere” in measure theory.

I'm slightly confused about the term almost everywhere as it is used in Folland's real analysis. Given a measure space $(X, \mathcal{M}, \mu)$ Suppose $f \equiv g$, $\mu$-almost everywhere where $f, ...
1
vote
1answer
85 views

Measurable vector bundles trivial

I hope you can help: If $E$ is a measurable vector bundle over a compact metric space $(X,\mu)$ then there is a subset $Y\subset X$ such that $\mu(Y)=1$ and $\pi ^{-1}(Y)$ is isomorphic to a trivial ...
2
votes
1answer
39 views

Does the sigma algebra generated by points include intervals?

is $[0,1] \in \sigma \left(\{\{x\} : x \in \mathbb{R}\}\right)$? I feel the answer should be no.
1
vote
0answers
35 views

$f_n\text{ bounded }, f_n\to f\text{ uniformly}$, then $f$ bounded

Let $(X,\mathfrak{A},\mu)$ be a measurable space, $f_n,n\geqslant 1$ measurable and bounded functions with $f_n\to f$ uniformly. Would like to know if then $f$ is measurable and bounded, too. ...
0
votes
1answer
188 views

f a real, continuous function, is it measurable?

Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function. I need to show that is a measurable function. I tried working with the definition: Let $f: X \to \mathbb{R}$ be a function. If ...
0
votes
1answer
135 views

Indicator function of a level set

Consider a positive definite locally Lipschitz function $V:\mathbb{R}^2\to\mathbb{R}_{\geq0}$. Fix $c\in\mathbb{R}_{\geq0}$ and consider the sublevel-set $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, ...
0
votes
0answers
40 views

Why Steiner Symmetrization makes a measurable set to a measurable one?

I find the Steiner Symmetrization is very useful in proving that the Hausdorff measure coincide with Lebesgue in the Euclidean space. However, I never saw anybody mention that the Steiner ...
1
vote
1answer
559 views

Smooth image of a null set has measure zero

Currently I'm looking at the proof of Sard's theorem given in John Milnor's "Topology from the differentiable viewpoint". I'm a bit confused about a remark on the case where the dimension of the ...
2
votes
1answer
71 views

Reference for studying polar coordinate

There is a theorem about justification of polar-coordinate in Folland-Real analysis p.78. I find it somewhat terse (Maybe it's just me).. I guess this kind of transform is possible even when ...
1
vote
1answer
77 views

A problem on a bounded and integrable function

Let $f$ be a bounded and integrable function. Define $g(h) = \int_{\Bbb R} |f(x+h) - f(x)|dx$. Prove that $\lim_{h \to 0}g(h) = 0$ I tried to approximate $f$ using bounded simple functions, but I ...
4
votes
2answers
126 views

Analysis/Inequality question about proving an infinite product greater than 0

This is from David Williams' book Probability using Martingales. I'm self-studying. Question Prove that if $$0\leq p_n < 1 \quad\text{ and }\quad S:=\sum p_n < \infty$$ then $$\prod (1-p_n) ...
0
votes
1answer
57 views

Need help to complete the proof about change of probability

Got a question about change of probability. $P$ and $Q$ are two probability measures on the same space $(\Omega,\Lambda)$,and let $f=\dfrac{dQ}{dP} $ denote the Randon-Nikodym derivative of $Q$ ...
4
votes
1answer
204 views

Isometry vs. measure preserving?

Consider functions between two measured metric spaces. What is the relation between an isometry and a function which preserves the measure of subsets? This question arose in my head as I thought ...
1
vote
1answer
64 views

Measurability on subsets.

Given a measure space $(X, \mathcal{M})$ we say that $f : X \to \mathbb{R}$ is $\mathcal{M}$-measurable if $f$ is ($\mathcal{M}, \mathcal{B}_\mathbb{R}$) measurable where $\mathcal{B}_\mathbb{R}$ is ...
1
vote
0answers
59 views

Finite convergence of integral over a space implies convergence of integral over all measurable functions in space.

I'm trying to prove the following: Suppose $\{f_n\}\in L^+$, $f_n \to f$ pointwise, and $\int f = \lim \int f_n < \infty$. Then $\int_Ef = \lim \int_E f_n$ for all $E$ in $\mathcal M$. $L^+$ ...
1
vote
1answer
71 views

Show: $X_n\xrightarrow{\mathcal{d}} X$, then $\mathbb{E}\lvert X\rvert\leqslant\liminf_{n\to\infty}\mathbb{E}\lvert X_n\rvert$

Let $X_n, X$ be random variables with $X_n\xrightarrow{d} X$. Show that then $$ \mathbb{E}\lvert X\rvert\leqslant\liminf_n \mathbb{E}\lvert X_n\rvert. $$ So let $X_n\xrightarrow{d} ...
2
votes
1answer
34 views

Show: $X_n\to X\text{ a.s.}\implies X_n\to X\text{ stochastically }\implies X_n\to X\text{ in distribution}$

Let $X_1,X_2,X_3,\ldots$ be random variables on one probability space $(\Omega,\mathcal{A},P)$. Show $$ X_n\to X\text{ a.s.}\implies X_n\to X\text{ stochastically }\implies X_n\to X\text{ ...
0
votes
1answer
721 views

Cardinality Of Borel Sets

I was trying to show that Borel $\sigma$ algebra is smaller than lebesgue measurable sets. I could come up with a proof for the cardinality of lebesgue measurable sets being $2^c$. Cardinality of ...
8
votes
1answer
585 views

Concentration of measure vs large deviation

When reading some probability publications I am always not sure why they call this or that inequality a 'concentration inequality' or 'large deviation inequality'. For me these (concentration of ...
1
vote
1answer
277 views

On union and intersection of non Lebesgue measurable sets

We know that union and intersection of two Lebesgue mble sets are also Lebesgue mble. My questions is about the same thing for non mble sets. 1. Does there exist two non mble sets(whose union is not ...
0
votes
1answer
21 views

Expectation of a stochastic integral conditioned on a particular σ-algebra

Suppose that $g$ is a simple process in the class $\mathcal{V}=\mathcal{V}[U,T]$. Using the notations $g_k=g(t_k)$, $\Delta B_k = B(t_{k+1})-B(t_k)$, and $\mathcal{F}_k=\mathcal{F}_{t_k}$, with the ...
0
votes
2answers
377 views

Formal definition of a random variable

I'm not new to the concept of random variable and I know the measure theory. Anyway, I started reading the book "Stochastic Differential Equation" by B. Oksendal, and I'm having some problem in ...
1
vote
1answer
359 views

Proof that the predictable sigma algebra is also generated by continuous and adapted processes

I'm reading George Lowther's blog and have a question about the proof of lemma 2. We want to verify that the predictable sigma algebra is also generated by the continuous and adapted processes. One ...
5
votes
1answer
193 views

Weakly convergent distributions on $\mathbb{R}$ with densities relating to Lebesgue-measure that do not converge

Show that there exist weakly convergent distributions on $\mathbb{R}$ that have a density relating to the Lebesgue measure $\lambda$ but the densities do not converge. Hint: ...
4
votes
3answers
269 views

Does $f(x)\in L^1$ imply that $f'(x) \in L^1$?

Let $f(x)$ be defined for all real numbers differentiable function of one variable.We know that: $$\int_{-\infty }^{+\infty } |f(x)| \, dx\neq +\infty$$ Problem is to resolve if it is possible or not ...
4
votes
1answer
60 views

Whether a set is closed or not

Denote by $C_{[0,1]}$ the ternary Cantor set on $[0,1]$. Now consider $[0,1] \setminus C_{[0,1]}$. It contains open intervals. Now define Cantor sets on all these open intervals by simply translating ...
1
vote
1answer
76 views

$\lim_{h\downarrow 0}\frac{1}{h}\int_{t-h}^tf(s)ds=f(t)$

Let $f:\mathbb{R}\to\mathbb{R}$ be Borel and bounded, then I was able to prove that the map $t\mapsto \int_{t-h}^tf(s)ds$ is Lipschitz continuous. Now if we assume in addition that $f$ is ...
2
votes
1answer
278 views

How does the natural filtration of a Brownian motion look like?

I am trying to understand how the natural filtration for a Brownian motion might look like. Definitions: I will start with the definitions for reference. The definition of a natural filtration is ...
22
votes
2answers
982 views

Lebesgue density strictly between 0 and 1

I am having trouble with the following problem: Let $A\subseteq \mathbb{R}$ be measurable, with $\mu(A)>0$ and $\mu(\mathbb{R}\backslash A)>0$. Then how do I show that there exists $x\in ...
4
votes
1answer
68 views

What keeps measure-preserving transformations from concentrating in a particular portion of a probability space?

I'm trying to show that for an event A with positive probability there is some n bounded by 1/P(A) such that $P(A \cap$ T$^{-n}A) > 0$, where T is a probability-preserving transformation. I'm ...
1
vote
1answer
58 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 ...
1
vote
1answer
75 views

Finding an example where a measure is not unique

Let $(X, \mathcal{M})$ be a measurable space. Let $\mu$, $\nu$ be measures defined on $\mathcal{M}$. (a) For $A \in \mathcal{M}$ define $\lambda(A)=\mu(A)+ \nu(A)$. Prove that $\lambda$ is a ...
1
vote
0answers
100 views

Norm of multiplication operator

I have that $(X,\Omega,\mu)$ is a sigma finite space, and I have that $g$ is a measurable function. Assume that $fg\in L^p$ for all $1\leq p\leq \infty$. I want to show that $g\in L^\infty$. My idea ...
3
votes
2answers
99 views

Using continuity of the measure in a proof…

I'm trying to understand the following proof: I don't understand how the conclusion came from the equation in the green box, did they use continuity of the measure?
2
votes
1answer
127 views

Intuition behind the weight function

The inner product in a $L^2$ space can be defined as: $$\langle f,g\rangle =\int_a^b \bar{f}(x)g(x)w(x)dx$$ For Legendre polynomials, we define it as: $$\langle P_m,P_n\rangle =\int_0^1 ...
3
votes
1answer
411 views

Understanding Lebesgue Integration

I have started studying Lebesgue integration and I have a few of questions regarding the Lebesgue integral: In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: ...
2
votes
0answers
146 views

Show $\pi$ is a measure

Show that $\pi(E)=sup\lbrace \mu(A): A\subseteq E, A\in\mathbf{X} \rbrace$ is a measure on $\mathbf{X}$. $\mu$ is a charge on $\mathbf{X}$ ($\sigma$-algebra), let $\pi$ be defined for ...
0
votes
1answer
42 views

Showing that $\int_{[0,1]}\frac{1}{x}\, d\lambda(x)$ isn't finite

$\lambda$ Lebesgue-measure on compact unit intervall, $\mu(x):=\frac{1}{x}\lambda(x)$. My question is how I can compute the integral $$ \mu([0,1])=\int 1_{[0,1]}\, ...
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vote
2answers
238 views

Showing that a function is Lebesgue integrable (or not)

How would I go about showing that the function $ f(x) = \left\{ \begin{array}{lr} &1 & : x \in [n,n+1) & :n\quad \text{even}\\ &-1 & : x \in [n,n+1) & ...
1
vote
1answer
106 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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vote
1answer
86 views

Sigma Algebra: Etymology

Why do we talk of sigma algebras in measure theory. As far as I know sigma is related to the countability. But what does it stand for?
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1answer
558 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 ...
0
votes
1answer
70 views

Every Cauchy sequence in $C([0,1])$ in the $L^2$ norm is also Cauchy in the $L^1$ norm.

I am asked to show the following: Proposition. There is a unique injection $j: L^2([0, 1]) \hookrightarrow L^1([0,1])$ which continuously extends $Id: C([0, 1]) \to C([0, 1])$. Here $L^1([0, ...
0
votes
1answer
138 views

Functions Lebesgue integrable over set

I have the following question State whether the function $f$ is Lebesgue measurable over $E$. Justify your answers, and calculate $\int_E f$ in those cases where this is feasible. i) $E = ...
-1
votes
1answer
50 views

Does $f<\infty$ a.s. imply that $f$ is integrable?

Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $f\colon\Omega\to\overline{\mathbb{R}}$ measurable. Does then $f<\infty$ a.s. imply that $f$ is integrable? I think no, but ...
0
votes
1answer
55 views

Confusion on a integration problem

Is it correct to ask to evaluate the integral: $$\int_{-1}^{1}\dfrac{1}{x}dx$$ The function $f(x)=\dfrac{1}{x}$ is not defined at $0$.
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0answers
25 views

How to go from generator to $\sigma$-field in this case?

Here is the problem: $\mathcal{S}\subset\mathcal{P}(\mathbb R)$ has the following property: $$\forall n\in\mathbb N \ \ \ \forall E\in\mathcal{S}: \ \ \ (n+E)\in \mathcal{S} $$ Show that ...
4
votes
0answers
196 views

Why do people apply Fubini-Tonelli theorem so easily?

I'm reading a text "Lebesgue Integration - Frank jones" from which i got recommended here, stackexchage. This text seemingly covers various topics on measure theory, but i think that's it. This text ...
1
vote
1answer
71 views

How do i prove this mixture of Lebesgue measurable functions is Lebesgue measurable?

Let $C:\mathbb{R}^n\times\mathbb{R}\rightarrow \mathbb{R}^{n+1}:((x_1,\cdots,x_n),x_{n+1})\mapsto (x_1,\cdots,x_{n+1})$ Let $f:\mathbb{R}^{n+1}\rightarrow [0,\infty]$ be a Lebesgue measurable ...