Every $mathbb{P}-$integrable function $uinmathcal{L}^1(mathbb{P})$ is bounded (Counterexample)

Mathematics Asked by Xenusi on November 21, 2020

Let $$(Ω,mathcal{A},mathbb{P})$$ be a probability space. Find a counterexample to the claim that every $$mathbb{P}-$$integrable function $$uinmathcal{L}^1(mathbb{P})$$ is bounded.

Countableexample

$$(Ω,mathcal{A},mathbb{P}) = (mathbb{R},mathcal{B}(mathbb{R}),delta_n)$$

$$delta_n$$ is the Dirac measure in point $$ninBbb N$$.

The function I have constructed is $$u=sum_{n=1}^infty n mathbb{1}_{A_n}$$. $$mathbb{1}_{A_n}$$ is the indicator function and $$A_n=[-n,n]$$. $$u$$ is unbounded, but have is it integrable?

On $$mathbb N$$, consider the probability measure $$mathbb P$$ defined by $$mathbb P({n}) = p_n propto frac 1{n^3}.$$

The identity function $$f:mathbb Ntomathbb R$$ given by $$f(n) = n$$ is not bounded and $$int_{mathbb N}|f|dmathbb P propto sum_{n=1}^infty frac 1{n^2} < infty.$$

Another counterexample could be the expectation of a random variable following a Poisson distribution.

Correct answer by volJunkie on November 21, 2020

Related Questions

Constant invariant quantities

0  Asked on November 6, 2021

Convergence of martingales is a martingale

1  Asked on November 6, 2021 by satana

Question on cardinality of a set

0  Asked on November 6, 2021 by rosco

$X$ is a random variable with PDF $f$, with $f$ continuous at $x=a$. Then $lim_{varepsilonto0^-}frac{1}{varepsilon}P(Xin(a,a+varepsilon))=f(a)$

0  Asked on November 6, 2021 by stackman

Suppose that $N$ and $r$ are positive integers. Prove or disprove that if $N$ is an even integer and $r$ is odd, then $binom{N}{r}$ is even.

3  Asked on November 2, 2021 by s1mple

Confusions about the proof of representations of isometries as products of reflections.

1  Asked on November 2, 2021 by user806566

Combinatorics doubts

1  Asked on November 2, 2021 by pietro-morri

$limlimits_{n rightarrow infty} e^{-2n}sum_{k=0}^n frac{(2n)^k}{k!}$

2  Asked on November 2, 2021

Why is $[1,2]$ relatively open in $[1,2] cup [3,4]$?

1  Asked on November 2, 2021

Algebra – How can I graph this function and work with exponential function with base $a$?

1  Asked on November 2, 2021

Best approximation of sum of unit vectors by a smaller subset

2  Asked on November 2, 2021 by g-g

Unbounded on every interval except null set but finite a.e

1  Asked on November 2, 2021

Jordan normal form of $;begin{pmatrix}0 & 1 & 0 \ 0 & 0 & 1 \ 0 & a & b end{pmatrix},; a,binmathbb{R}$

1  Asked on November 2, 2021 by user731634

Derivative of random normal times indicator function

1  Asked on November 2, 2021 by forumwhiner

Critical points of a nonnegative quadratic form on a subspace

1  Asked on November 2, 2021

Geometric interpretation of a symmetric matrix

0  Asked on November 2, 2021

Minimum number of solutions for $f=g$

0  Asked on November 2, 2021

$X_i equiv a_i pmod{P}$ for some $a_i in mathcal{O}$ given a prime ideal $P$ of $mathcal{O}[X_1, ldots, X_n]/(f_1, …, f_n)$

1  Asked on November 2, 2021

A limit involving the integer nearest to $n$-th power

2  Asked on November 2, 2021 by tong_nor

Proving that $F(x)=sumlimits_{k=1}^{p-1}left(frac{k}{p}right)x^k$ has at least $frac{p-1}{2}$ different complex roots

0  Asked on November 2, 2021 by geromty