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

Question

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?

volJunkie · Accepted Answer

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.

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

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