Looking for counter example where monotone convergence sequence not hold for a decreasing sequence

Question

Let $\{f_n\}$ be an increasing sequence of nonnegative measurable functions on $E$. The monotone convergence sequence then states that:

If $f_n \rightarrow f$ pointwise almost everyhwere on $E$, then $\lim_{n \rightarrow \infty} \int_E f_n = \int_E f$.

Why does this theorem not hold for decreasing sequences of functions? I'm baffled why it wouldn't, since I feel like everything would be symmetric in a way or whatever... Does anyone have an explicit counter example? Thanks!

Answer 1

Take the sequence $f_n(x)=\frac{x}{n}$ on $[0,+\infty)$