Question: Consider on the measure space (left([0, infty), mathscr{B}[0, infty),left.lambda^{1}ight|_{[0, infty)}ight)) the filtration defined by (mathscr{A}_{n}:=sigma([0,1),[0,2), ldots,[n-1, n),[n, infty))). Find (mathbb{E}^{mathscr{A}_{n}} u) for (u in L^{p}).
Consider on the measure space \(\left([0, \infty), \mathscr{B}[0, \infty),\left.\lambda^{1}ight|_{[0, \infty)}ight)\) the filtration defined by \(\mathscr{A}_{n}:=\sigma([0,1),[0,2), \ldots,[n-1, n),[n, \infty))\). Find \(\mathbb{E}^{\mathscr{A}_{n}} u\) for \(u \in L^{p}\).
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