Approximation of integrable andom variables by square integrable random variables

Question

Let W be a Brownian Motion and $ \mathcal{F}$ be the augmented Filtration which is generated by the Brownian Motion. Y be a $ \mathcal{F}(T)$ measurable integrable Random Variable. In this Situation we can approximate Y by square integrable random variables $Y_{k}$ such that$\lim\limits_{k \rightarrow \infty}$ $\lVert Y-Y_{k} \rVert_{L_{1}(P)}$=0. Why is this possible ? (This is used to proof Martingale Represation)

Answer 1

Let $Y_k := Y 1_{|Y| < k}$. Since each $Y_k$ is bounded, they are clearly in $L^2$. Then because $\mathbb{E}[|Y|] < \infty$, we have $|Y| < \infty$ a.s. so $(Y_k) \rightarrow Y$ almost surely. Finally, $|Y_k| \le |Y|$, so by the dominated convergence theorem $(Y_k) \rightarrow Y$ in $L^1$ as desireed.