1
vote
1answer
56 views

Determining whether an uncountable set of integral equations yield a unique solution

I am interested in the set of numbers $\alpha>0$ for which there exists a function $g:\mathbb{R}\to[0,1]$ satisfying $$ \forall r\in \mathbb{R} \qquad f(r) = \int\limits_\mathbb{R}\! g(\alpha ...
2
votes
1answer
29 views

What is the proper definition of cylinder sets?

in class we defined the terminal $\sigma$-algebra for a sequence of random variables $(X_i)$ with $X_i:\Omega \rightarrow \mathbb{R}$ as $G_{\infty}:=\bigcap_i G_i$, with ...
2
votes
3answers
273 views

Integral change that I don't understand

If $f(t)$ is a Probability density function of a positive RV. $\int_0^\infty\int_x^{\infty}f(t)dtdx$ Using fubini theorem should become $\int_0^\infty\int_0^{t}f(t)dxdt$ But why? Surely the answer ...
4
votes
1answer
39 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
0
votes
1answer
76 views

Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

In one of the proofs, my text mentions that if $f,g$ are Lebesgue integrable then $fg$ is integrable with respect to a probability measure. I guess I have missed something, since it doesn't look ...
2
votes
0answers
43 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
2
votes
1answer
66 views

Convergence almost sure pointless?

A very common type of convergence in probability theory is 'almost sure convergence'. I don't understand why this type is used at all. In principle, we should always be able to substitute it by a ...
1
vote
1answer
47 views

Integrate over different measures

In Probability theory the expected value of a random variables $X : \Omega \rightarrow \mathbb{R}$ is defined as $E(X) = \int_\Omega X dP$ Now, if $\Omega \subset \mathbb{R}$ and has a density ...
1
vote
0answers
37 views

Not measurable function whose module is measurable

I read through my notes that is trivial to find a not measurable function $f$ whose module $|f|$ is measurable. However I don't know how to provide such an example.
1
vote
0answers
39 views

Right-continuity of functions associated to measures

I would like to show that it's possible to associate to a measure a monotone increasing right-continuous function s.t.: $\mu(\left(a,b\right])=F(b)-F(a)$. How can I prove that a function like ...
1
vote
3answers
77 views

A basic question on integration [closed]

$x^{k}{\rm e}^{-x^{2}}$ decreases to zero "exponentially" when $x \to \pm \infty$, $\int_{\mathbb R}{\rm f}\left(x\right)\,{\rm d}x < \infty$. Which theorem is being used here ?
0
votes
1answer
30 views

Integrate function with image $\mathbb{R}^n$

I know that for any measure space $(\Omega,\Sigma,\mu)$ and any $\Sigma$-borel-measurble function $f\colon \Omega \to \mathbb{R}$ the integral $$\int_\Omega |f(x)| \, d\mu(x)$$ is well definied. I ...
3
votes
1answer
105 views

Probabilistic Proof That An Absolutely Continuous Function is Differentiable Almost Everywhere

Consider the probability space $([0,1), \mathcal{B}, \lambda)$ where $\mathcal{B}$ is the Borel $\sigma$-algebra and $\lambda$ is the uniform measure. Let $A_{i,n} = [(i-1)2^{-n}, i2^{-n})$ for $i \in ...
2
votes
1answer
121 views

Notation used for integrals w.r.t. probability measures

I was staring at one of my questions at SE and realized that I do not really understand what I mean by $dP(\omega)$ when I write: $$EX = \int_{\Omega} X(\omega) \, dP (\omega)$$ where $X: \Omega \to ...
0
votes
1answer
54 views

Sufficient and necessary condition for Lebesgue integrability of a random variable

Could anyone give me a hand with the following problem? Let $f$ be a random variable over a probability space $(\Omega,A,\mathbb P)$. Show that $f$ is integrable $\iff $ ...
0
votes
3answers
181 views

Why is expectation defined by $\int xf(x)dx$?

I recently found out that the expectation of a random variable $X$ in a probability space $(\Omega, \mathcal F, \mathbb P)$, $\mathbb E(X)$, is just the term used in probability theory for the ...
2
votes
2answers
118 views

Integral of Schwartz function over probability measure

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$ such that ...
2
votes
1answer
39 views

Integral of exponent of random variable is continuous

Let $X$ be a set, $F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given a random variable $f:X\rightarrow\mathbb{R}$, define $$\chi_f(t)=\int_Xe^{itf}d\mu$$ Show ...
4
votes
2answers
297 views

Need help badly: n-dimensional Lebesgue measure of a hyperplane is zero.

Let $\alpha \in \mathbb{R}$, $a \neq 0$, and $\mu \in \mathbb{R}^{n}$. Let $H$ be the hyperplane in $R^{n}$ given by $h = \{ x \in \mathbb{R}^{n} : \langle x-\mu , a \rangle = 0 \}$. Show that ...
1
vote
1answer
53 views

Proving that some truncations have the same expected value (using Lebesgue integral)

Let $X_1,X_2\colon(\Omega,\mathcal F,\mathbb P)\to (\mathbb R,\mathcal B(\mathbb R))$ be two random real variables, where $(\Omega,\mathcal F,\mathbb P)$ denotes a probability space. And $(\mathbb ...
0
votes
2answers
67 views

Expectation of nonnegative RV

I'm taking an intro course in Probability theory, and we have just defined expectation for a random variable as $E(X) = E(X^+) - E(X^-)$ if either of them is finite (extending the definition first ...
1
vote
0answers
89 views

On continuity of measure

Let $m$ be a probability measure on $\mathbb{R}^n$. Consider a function $\ f: \mathbb{R}^n \rightarrow \mathbb{R}$. Say under what conditions the following inequality holds. $$ m\left(\left\{ x \in ...
1
vote
1answer
134 views

Double expected value

Let $m$ be a probability measure on $\mathbb{R}^n$, so that $m(\mathbb{R}^n) = 1$. Consider two measurable functions $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$, and $g : ...
1
vote
2answers
69 views

Convergence of $\int_{A_n} f$ to $0$

I am looking for a name or a reference in a textbook for the following result in order to quote it. For any $f\in L^1(\mathbb{R})$-integrable function, we have $$\lim_{n\to\infty}\int_{A_n} ...
2
votes
1answer
53 views

Continuity of a probability integral

Let $m: \mathcal{B}(\mathbb{R}^n) \rightarrow [0,1] $ be a probability measure without point masses. Let $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be (jointly) continuous. Define ...
1
vote
1answer
104 views

Continuity of a probability integral with supremum

Let $m: \mathcal{B}(\mathbb{R}^n) \rightarrow [0,1] $ be a probability measure without point masses and let $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be continuous. Let $\epsilon ...
8
votes
1answer
209 views

$L^{p}$ functions from Rudin Exercises 3.5

I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
1
vote
1answer
90 views

Question about integration (related to uniform integrability)

Consider a probability space $( \Omega, \Sigma, \mu) $ (we could also consider a general measure space). Suppose $f: \Omega -> \mathbb{R}$ is integrable. Does this mean that $ \int |f| \chi(|f| ...
1
vote
1answer
67 views

Converging almost surely and value of the integral

Given a sequence of nonnegative functions $(f_n)_n$ converging almost surely (for the Lebesgue measure $d\mu$) to $f$. Assume that $\int f_n d\mu \rightarrow c < \infty$ as $n$ goes to infinity. ...
1
vote
1answer
140 views

Uniform integrability for a single random variable

Let $X$ be a random variable. Are the following three equivalent? $X \in L^1$, i.e. $E |X| < \infty$. $X$ is uniformly integrable. That is, if given $\epsilon>0$, there exists $K\in[0,\infty)$ ...
5
votes
1answer
206 views

Equi-integrability of a single function: is it the same as summability?

Let $(\Omega, \mathcal{M}, \mu)$ be a measure space and let $f\ge 0$ be a measurable function on $\Omega$. Suppose that $f$ satisfies the following properties: For all $\varepsilon > 0$ there ...
0
votes
1answer
100 views

The Lebesgue integral $\int_\Omega dP$

I am a beginner. Given probability measure $P$ and sample space $\Omega$, is it true that: $$\displaystyle \ \ \int_\Omega dP = 1$$
2
votes
1answer
121 views

Uniform Integrability after composition

Let $\mu$ be a finite measure on $X \subseteq \mathbb{R}^n$. Consider the Uniformly Integrable family $\{ f_n(\cdot) \}_{n \in \mathbb{N}}$ of functions $f_n : X \rightarrow \mathbb{R}_{\geq 0}$. ...
5
votes
0answers
1k views

Dunford-Pettis Theorem

The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that: A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact. Now ...
4
votes
1answer
385 views

Uniform Integrability

Let $\mu$ be a probability measure on $X$, so that $\int_X \mu(dx) = 1$. I have a family $\{f_i\}_{i=1}^{\infty}$ of functions $f_i: X \rightarrow \mathbb{R}_{\geq 0}$ such that $$ \displaystyle ...
25
votes
8answers
4k views

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...