Almost sure boundedness is a strong condition which implies the finiteness of moment at any order. Now let $f:\mathbb R \to \mathbb R$ be square-integrable (w.r.t. Lebesgue measure), i.e., $\int f^2(x) \mathrm d x < \infty$. Could you provide an example of such $f$ that is not almost surely bounded (w.r.t. Lebesgue measure)?
Example of a square-integrable function that is not almost surely bounded
measure-theoryreal-analysis
Best Answer
$\frac{1}{\sqrt{x}}$ in $(0,1)$ with the Lebesgue measure is a classic example in showing that $L^{1}$ need not be a subset of $L^{\infty}$ . (in fact $L^{1}$ need not be a subset of $L^{p}$ for any $p>1$.
So just take the square root (i.e. $\frac{1}{x^{\frac{1}{4}}})$ and you'll get an $L^{2}$ function that is not $L^{\infty}$