Integrability condition

Question

Suppose that \begin{align} \mathbb{E}\int_{0}^{T}f^{2}(t)dt <K \end{align} Does it also hold that \begin{align} \int_{0}^{T}f^{2}(t)dt <K \end{align} ? Here, T, K>0 are fixed. I am a bit confused !

Answer 1

Here are some remarks. Let $X$ denote any nonnegative random variable.

• If $E(X)$ is finite then $P(X\lt\infty)=1$.
  
  Proof: $[X=\infty]\subseteq[X\geqslant n]$ for every $n$ and $P(X\geqslant n)\leqslant E(X)/n$ by Markov inequality, QED.

• On the other hand, $E(X)\leqslant K$ with $K\gt0$ does not imply that $P(X\leqslant K)=1$. Counterexample: $X$ uniform on $(0,2K)$.

• Finally, the statement "$X\leqslant K$" can only mean that $[X\leqslant K]=\Omega$ or that $P(X\leqslant K)=1$. The latter is usually referred to as "$X\leqslant K$ almost surely".

Now, apply these remarks to the nonnegative random variable $X=\displaystyle\int_0^Tf^2(t)\,\mathrm dt$.