0
votes
1answer
29 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
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
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.
3
votes
1answer
90 views

Exercise on measure theory

Let $X\neq \emptyset$ and $f:X \rightarrow [0, \infty]$ not identical infinity. Set $$ \sum_{x \in X} f(x)= \sup \left\{ \sum_{x \in F}f(x), F \subseteq X, F \mbox{ finite} \right\}.$$ $(i)$ Show ...
0
votes
1answer
53 views

Continuity of a function defined by means of the Lebesgue measure

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue measurable function and $\phi(x)=\lambda ( \lbrace{ t: f(t) >x \rbrace} )$. Prove that $\phi$ is right-continuous but not necessarily ...
1
vote
0answers
40 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
1answer
93 views

Lebesgue integral of a bounded random variable

Given a random variable $X$, if we take a measurable and bounded function $f(X)$ then can we say that $f$ is Lebesgue integrable wrt a probability measure on $\mathbb R$? In Real Analyses book by ...
0
votes
1answer
29 views

Does this inequality hold for a Lipschitzian strictly decreasing function?

There is a function $f : [0;1]^n\rightarrow \mathbb{R}$ which is Lipschitzian and strictly decreasing in all variables. Is it possible to prove either one of these two statements? There is an $M > ...
2
votes
1answer
84 views

integral over almost sure existing derivatives

Let $f$ and $g$ and $f-g$ be real-valued, Lipschitz functions, with Lipschitz constant smaller or equal to 1, on $[a,b]$ whose derivatives are positive and exist only $\lambda$-almost surely. Does ...
2
votes
1answer
137 views

Monotone convergence

Consider $X$ as non-decreasing non-negative function. Consider $\mu$ and $\nu$ as two probability measures on $(\mathbb{R},\mathcal{B})$ for which we know $\mu([t, \infty)) \geqslant \nu([t, \infty)) ...
2
votes
2answers
101 views

Expectation and Lebesgue integration question

How can I show: If a random variable $Z$ has finite expectation $E(Z)$ (i.e., $Z$ is Lebesgue integrable), then $nP(|Z|>n) \to 0$ as $n \to \infty$?
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 ...