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

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4
votes
2answers
195 views

Minimizing the expectation over a set, wrt to the Gaussian measure

I have recently read a proof [1] where, at the last step, the authors use an inequality which basically amounts to a lower bound on $\int_\mathbb{R} \mathbf{1}_A(x)|x| \phi(x)dx$, where $\phi$ is the ...
0
votes
1answer
79 views

Computing outer measure

Compute $m^*(\{(1+\frac{1}{n})^n:n\in N\})$ I'm fairly new to outer measures and having trouble using the definition of an outer measure to compute this. Thank you for any help!
0
votes
1answer
30 views

If $P$ is a statistically complete set of distributions, the only sufficient subfield is the trivial one

In this thread i solved a claim stated without proof by Bahadur that if $P=\left\{p\right\}$ is the set of all probability measures on the measurable space $\left(\Omega,\mathcal{A}\right)$, ...
0
votes
1answer
186 views

Outer Measure of a Finite Covering of the Rationals on $[0, 1]$

I'm studying for my Real Analysis final and came upon an old question on outer measure that I'm pretty sure I'm doing wrong. If $B$ is the set containing the rationals on $[0, 1]$, and ...
1
vote
1answer
136 views

$f$ integrable $\implies g(x) = \int_{-\infty}^x f$ is absolutely continuous

Suppose that $f : \mathbb{R} \to \overline{\mathbb{R}}$ is an integrable function. Show that the function $g : \mathbb{R} \to \mathbb{R}$ defined by $g(x) = \int_{-\infty}^x f$ is absolutely ...
1
vote
1answer
166 views

Tail events and exchangeable events

In this problem I have $X_1, X_2, \cdots$ independent identically distributed RVs taking values $\pm1$ with the equal probability of $1/2$ and my trajectory is defined by $S_n=\sum_i^n X_i$ (so pretty ...
2
votes
1answer
74 views

Can I deal with the weak derivative in the “strong” sense?

This is an exercise in functional analysis: For $k=1,2,3$, let $A_k: D(A_k)\subset L^2([0,1])\to L^2[(0,1)]$ be the first-order differential operators $A_ku=iu'$ with domains $$ D(A_1) = ...
3
votes
2answers
289 views

Why this is not a sigma algebra

Let be $\Omega$ the interval $(0,1]$ and let $\mathcal{F}$ be the set of all sets of the form $(a_0,a_1]\cup(a_2,a_3]\cup\cdots\cup(a_{n-1},a_n]$, where $0\le a_0\le a_1\le\cdots\le a_n\le 1$. Show ...
1
vote
2answers
56 views

Question in Lebesgue integrable functions.

Suppose $g$ be a measurable function satisfying: $∀$ $σ∈[c,d]$ , there exists $δ>0$ such that $∫_E|g| <∞$ where $E=[σ-δ, σ+δ]$. Prove that $g$ is Lebesgue integrable on $[c, d]$.
3
votes
0answers
92 views

A question about the stability of a property of the normal distribution

Recall that the standard normal distribution can be characterized as the unique standardized (having mean zero and unit variance) distribution $P$ on $\mathbb{R}$ with the property that with $X$, $Y$ ...
0
votes
1answer
582 views

Monotone increasing sequence of random variable that converge in probability implies convergence almost surely

Let $\{X_n\}$ be a collection of random variable with $X_{n+1} \geq X_n$ for all $n$ and $X_n \rightarrow X$ in probabilty. How to prove that $X_n \rightarrow X$ almost surely. My partial answer: ...
2
votes
1answer
45 views

about well-defined integral kernel

Let $\phi:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ measurable function such that $$ \int_{\mathbb{R}^n}|\phi(x,y)|\ dx \leq M\ , \quad \int_{\mathbb{R}^n}|\phi(x,y)|\ dy \leq M\,.$$ Let $f\in ...
2
votes
1answer
99 views

Measure on Boolean algebra

my question is: Suppose that $\mathfrak{B}$ is a measurable Boolean algebra, does this mean that "Every measure on $\mathfrak{B}$ should be strictly positive ? or this will be the case after ...
2
votes
1answer
75 views

Combining convergence in probability and the means of the positive sequence of r.v. implies convergence in L 1

Let $\{X_n\}$ be a collection of positive random variable with $X_n \rightarrow X$ in probability. Prove that if $E(X_n) \rightarrow E(X)$, then $X_n \rightarrow X$ in $L^1$. My partial answer: Let ...
1
vote
2answers
88 views

Cardinality of the class of $G_\delta$ subset of $\mathbb{R}$ of Lebesgue measure zero

Let $\mathcal{N}$ be the class of all subsets of $\mathbb{R}$ of Lebesgue measure zero and let $\mathcal{G}_\delta$ be the class of all $G_\delta$ subsets of $\mathbb{R}$. How do I show that ...
3
votes
1answer
93 views

Measurability of a function defined on a product measure space, and related to a measurable function

Let $ (X,\mu) $ be a standard measure space - so that we may assume that $X$ is the unit interval $[0,1]$ with the Borel $\sigma$-algebra. Consider $X \times X$ with the product measure $\mu \times ...
0
votes
1answer
71 views

Definitions of K-L divergence based on likelihood ratio and on R-N derivative

From Wikipedia For distributions $P$ and $Q$ of a continuous random variable, KL-divergence is defined to be the integral: $$ D_{\mathrm{KL}}(P\|Q) = \int_{-\infty}^\infty ...
0
votes
1answer
57 views

Uniqueness of singular measure for inner function

A singular inner function $M$ (an analytic function on the open unit disk without zeros which takes on unimodular boundary values almost everywhere) can be written as $$M(z)=c \exp\left(\int_0^{2\pi} ...
3
votes
3answers
131 views

Why does the Lebesgue Outer Measure of $\mathbb{R}$ is $+\infty?$

If $A\subseteq \mathbb{R}$, then the Lebesgue outer measure of $A$, denoted $m^*(A)$, is defined to be $$m^*(A)=\operatorname{inf}\left\{\sum_{k=1}^{+\infty}\ell(I_k)\right\}$$ where the infimum is ...
1
vote
1answer
152 views

Approximating measurable functions on $[0,1]$ by smooth functions.

Let $f$ be a measurable function on $[0,1]$. Is there a sequence infinitely differentiable $f_n$ such that one of $f_n\rightarrow f$ pointwise $f_n\rightarrow f$ uniformly ...
2
votes
1answer
176 views

Measurable functions on product measures

Let $ (X,\mu) $ be a measure space, and consider $X \times X$ with the product measure $\mu \times \mu $. Consider two functions $f$ and $g$ defined on $X \times X$ such that: $f$ is measurable. For ...
3
votes
1answer
57 views

A (partial) argument converting sums in $\ell^1$ into Lebesgue-integrable functions.

First, I want to mention that this problem is off of a take home final I have. I was given permission to research/ask about this specific line of reasoning, in large because I think the professor ...
2
votes
2answers
145 views

Any example for a function having domain and range as subset of real line that is NOT Borel function?

Suppose there is a function $f:A\to B$ where $A,\,B\subseteq\mathbb{R}$, is there any example for this function being NOT Borel function? Well the question came up to be when I was reading the ...
1
vote
2answers
320 views

Outer measure proof, assuming measurable set exists

Let $A\subset X $ be a null set (so $m^*(A)=0$). Assume that $X:=[a,b]$ is a fixed interval in $\mathbb R$ and let $m^*$ be the outer measure of $X$. Show that $A\subset X$ is a measurable set if and ...
2
votes
1answer
149 views

Interchange differential operator with Lebesgue integral.

Under what condition am I able to interchange a differential operator with an integral? More precisely, given a function $f:\Omega\times U\to\Bbb R$ from a measure space $(\Omega,\mathscr A,\mu)$ and ...
0
votes
2answers
350 views

densities being absolutely continuous wrt Lebesgue measure

I'm reading an article with an assumption similar to: "The density $f(.)$ exists and is absolutely continuous with respect to Lebesgue measure". I don't understand this assumption because $f$ is not ...
4
votes
1answer
203 views

strict convexity with a measure theoretic property

Suppose $(x_n)$ is a positive sequence of reals converging to $x$. Furthermore we have a measure space $(E,\mathcal{E},\mu)$ given, with finite measure $\mu$. There are a measurables nonnegative and ...
0
votes
1answer
65 views

Probability of two events are indepedent

Given a probability space $(\Omega, \mathscr {B}, P)$, then $\sigma : \mathscr{B}\times \mathscr{B} \to [0,1]^2$ is defined as, for any $A, B \in \mathscr{B}$ $$(A,B) \mapsto (P(A),P(B))$$ Now take ...
5
votes
3answers
418 views

What measure does Lebesgue measure induce on the fat Cantor set?

I know that the fat Cantor set under the subspace topology is homeomorphic to Cantor space $\{0,1\}^{\mathbb N}$ under the product topology induced by the discrete topology on $\{0,1\}$. Call the ...
8
votes
1answer
489 views

Image of a set of zero measure has zero measure

I am studying for my final and got stuck on the following problem from the previous year. I put my attempt below. Suppose that $I\subset \mathbb{R}$ is an open interval, $f:I\rightarrow \mathbb{R}$ ...
1
vote
1answer
61 views

Extension of Fourier Transform

We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
0
votes
2answers
184 views

Strict inequality in Reverse Fatou lemma: $\varlimsup \int f_n\le \int \varlimsup f_n$

Let $\{f_n\}$ be a sequence of nonnegative functions dominated by some function $$ g \in L^1. $$ Then, the reverse Fatou lemma says $$ \limsup \int f_n \le \int \limsup f_n. $$ Is it possible to ...
4
votes
1answer
671 views

Interchange supremum and expectation

Let $B_n:=\{f\in L^\infty_+\mid f\le n \}$, where we consider $L^\infty$ with the weak$^*$ topology. I have the following sets $$D(z):=\{h\in L^0_+(\mathcal{F}_T)\mid h\le Z_T \mbox{ for a }Z\in ...
0
votes
1answer
163 views

Counterexamples for Borel-Cantelli

Our teacher mentioned to construct two counterexmaples for Borel-Cantelli using the following ways. (a) Construct an exmaple with $\sum_{i=1}^{\infty}\mathbb P(A_i)=\infty$ where $\mathbb ...
2
votes
1answer
128 views

Lebesgue Integral defined on infinite measure

Royden's Real Analysis Question: Let {$a_n$} be a sequence of nonnegative real numbers. Define the function $f$ on $E=[1,\infty)$ by setting $f(x)=a_n$ if $n\leq x< n+1$. I want to show that ...
0
votes
1answer
30 views

Computing $\mathbb{E}[X_1S^4]$

Given $X_1,X_2,\cdots$ i.i.d. random variables with $\mathbb{E}[X_i]=0$. If we are given $S=\sum_{i=1}^{10}X_i$ and the fact that $\mathbb{E}[S^5]=30$ what method do you need to compute ...
2
votes
1answer
250 views

Verifying Fatou's Lemma

Royden's Real Analysis Question: Let {$f_n$} be a sequence of nonnegative measurable functions on $R$ such that $f_n\implies f$ pointwise on $E$. Let $M\geq0$ be such that $\int_Ef_n\leq M$ for all ...
1
vote
0answers
121 views

Showing that a piecewise function is measurable

I'm doing an assignment for a (first) course in analysis, and I'm having some trouble with showing that functions are measurable. In this problem $(X,\mathcal{A},\mu)$ and ...
3
votes
2answers
487 views

Intersection of $\sigma$-algebras and set theory

Theorem: Given $\{E_{\alpha}\}_{\alpha \in \mathcal{A}}$, where each $E_\alpha$ is a $\sigma$-algebra on $X$. Then $E:=\bigcap_{\alpha \in \mathcal{A}}E_\alpha$ is a $\sigma$-algebra. Proof: Take ...
3
votes
1answer
71 views

Application of Strong Law of Large Numbers and Fubini's Theorem

This problem comes from here. I am not looking for help on solving the problem, actually to understand something said in the setup: Let $F$ be a distribution with $F(0-) =0 $ and$F(1)=1$. Let ...
1
vote
1answer
302 views

Measure theory properties proof

For a set $A \subset \mathbb R$, $\alpha \in \mathbb R$ and $x_o \in \mathbb R$, put $x_0 + A:=${$x_o+a:a \in A$} and $\alpha A:=${$\alpha a:a \in A$} Let $m^*$ be an outer measure on a set ...
0
votes
1answer
73 views

Absolute Convergence of a Function

I have got stuck with a question. Please help me. Prove that $\dfrac{\sin(x)}{x}$ belongs to $L^p$ for all $p>1$. Thank You.
1
vote
0answers
193 views

Are continuous functions strongly measurable?

Measure theory is still quite new to me, and I'm a bit confused about the following. Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and ...
2
votes
1answer
56 views

Concerning the noncompactness of the map, $I: W^{1, 2}(\mathbb{R}^n)\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$

Consider the identity map $I:W^{1,2}(\mathbb{R^n})\rightarrow L^{2^{\ast}}(\mathbb{R}^n)$ where $n\geq 3$. Suppose that this map is not compact that is given some bounded sequence of functions ...
2
votes
2answers
264 views

Interpretation of dP in Radon-Nikodym Theorem

[Radon-Nikodym Theorem] Let $(\Omega, \Sigma, P)$ be a probability space. Suppose that $(\Omega, \Sigma, \mu)$ be a measure space with $\mu(A)=0$ implies $P(A)=0$, then there exist a function $f:X ...
1
vote
0answers
153 views

Change of probability measure and a continuous-time Markov chain

Let $(\Omega,\mathcal{F},\mathbb{P},\mathbb{F})$ be a complete filtered probability space, with $W$ a Wiener process and $\alpha$ a continuous-time Markov chain (taking values in $\{1,...,M\}$). We ...
0
votes
2answers
177 views

Conditional Expectation of Exponential Order Statistic $\text{E}(X_{(2)} \mid X_{(1)}=r_1)$

Having already worked out the distributions of $\Delta_{(2)}X=X_{(2)}-X_{(1)}\sim\text{Exp}(\lambda)$ and of $\Delta_{(1)}X=X_{(1)}\sim\text{Exp}(2\lambda)$ where $X_{(i)}$ are the $i$th order ...
0
votes
1answer
250 views

A Borel measurable function which is not continuous

I want to find a example of Borel measurable function which is not continuous. I think that it is a simple or step function or semicontinuous function. Please help me for find it. Thanks.
2
votes
0answers
86 views

When $ A \int_0^{\infty} e^{-\lambda t}S(t)u dt = \int_0^{\infty} e^{-\lambda t}S(t)Au dt$?

I have real Banach space $X$ and a bounded linear operator $S: X \to X$ which satisfy: 1) $S(0)u = u$ $\text{ }$ for all $u \in X$ 2) $S(t+s)=S(t)S(s)u = S(s)S(t)u $ $\quad$($ t,s \geq 0$, $u \in X ...
3
votes
1answer
114 views

Does convergence in $L^p$ and pointwise imply the same limit?

If $f_n\in L^p$ converge to $f$ pointwise and to $g$ in $L^p$. Does that mean $f=g$ almost everywhere?