real analysis - $lim_{nrightarrow infty }frac{1}{n}int_{Y

$Y$ is a non-negative random variable, not necessary integrable. How to show
$$\lim_{n\rightarrow \infty }\frac{1}{n}\int_{Y

My idea is to find a good upper bound that converges to zero. However, I didn't find a good one. My upper bound is
$$\frac{1}{n}\int_{Y

Answer

Note that the sequence of random variables $X_n = \frac{1}{n} I\{Y < n\} Y$ is bounded by $1$ and converges pointwise to $0$. The assertion now follows from the dominated convergence theorem.