# 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

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