Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. Find a counterexample to the claim that every \(\mathbb{P}\)-integrable function \(u \in \mathcal{L}^{1}(\mathbb{P})\) is bounded.

[ you could try to take \(\Omega=(0,1), \mathbb{P}=\lambda^{1}\) and show that \(1 / \sqrt{x}\) is Lebesgue integrable on \((0,1)\) by finding a sequence of suitable simple functions that is below \(1 / \sqrt{x}\) on, say, \((1 / m, 1)\) and then let \(m ightarrow \infty\) using Beppo Levi's theorem.]