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Let $(\Omega, \mathcal F, P, \mathcal F_t)$ be a filtered probability space. Consider two spaces $M$ and $S$ defines as follows:

$M$ is a collection of all continuous $\mathcal F_t$-adapted processes of the form $\phi: [0,1]\times \Omega \mapsto \mathbb R$ satisfying $$\mathbb E [\int_0^1 \phi_s^2 ds]<\infty.$$

$S$ is a collection of all continuous $\mathcal F_t$-adapted processes of the form $\phi: [0,1]\times \Omega \mapsto \mathbb R$ satisfying $$\mathbb E [\sup_{0\le t \le 1} \phi_t^2]<\infty.$$

[Q.] It is obvious that $S\subset M$. Can we say $S = M$? If not, find a $\phi\in M\setminus S$.

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    $\begingroup$ I think you can consider the trivial case where the $\phi$ do not depend on $\omega$ at all. Then you just need to select a function in $L^2$ but not in $L^\infty$, for instance a function with a singularity like that of $x^{-1/4}$. $\endgroup$ – Ian Dec 5 '14 at 16:07
  • $\begingroup$ You are right. But, I meant to say $M$ and $S$ are subset of continuous path. In this case, we do not have deterministic path in $L^2\setminus L^\infty$. Original text is changed now. $\endgroup$ – user79963 Dec 6 '14 at 15:16

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