Let (left(B_{t}ight)_{t geqslant 0}) be a (mathrm{BM}^{1}). Show that ((0, infty) i s mapsto f(s, omega):=mathbb{1}_{(0, infty)}left(B_{s}ight))

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Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \((0, \infty) i s \mapsto f(s, \omega):=\mathbb{1}_{(0, \infty)}\left(B_{s}ight)\) is continuous in \(L^{2}(\mathbb{P})\).

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