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Ito Integral

  1. Consider a set of stochastic process $f(t)$ mainly such that

a) $$ E\left(\int_0^{+\infty}f(t)^2 \,dt\right) < \infty. $$

Denote this set of stochastic process as $M^2$.

Question:

a) For each $w$, $f(t,$$w$$)$ is a continuous function, so we all know a bounded continuous function will be Riemann integrable, but why it is square integrable too, i.e. why $$ \int_0^{\infty}f(t)^2 \,dt $$ has to exist

i.e. why the set $$ M^2 $$ will for sure be in $$ L^2 $$ when integrating over time $t$ on the time line $(0, \infty)$?

Thank you

2 Answers 2