Questions around the establish of Ito integral

Question

I got a some detailed questions on the Ito integral and hope someone can help.

1. I'm reading Chap 3 of Oksendal's SDE book. There he establishes the Ito integral and the Ito isometry for simple processes which is fine with me. But then he introduced 3 steps to show for any adapted $f(t,\omega)$ as long as it satisfies $\displaystyle E[\int_0^T f^2(t,\omega)dt]<\infty$ it can be approimated by a serie of simple process $(\phi_n)$ i.e. $\displaystyle \lim\limits_{n\to\infty}E(\int_0^T[\phi_n(t)-f(t)]^2dt)=0$.

   I can understand the 3 steps (simple process $\to$ bounded continuous process $\to$ bounded process $\to$ $f(t,\omega)$), but I failed to see how to link them together to give the ideal $\displaystyle \lim\limits_{n\to\infty}E(\int_0^T[\phi_n(t)-f(t)]dt)=0$. In particular, I think I need some uniform convergence property in the approimation in the 3 established steps to give the final result. Yet the prove in the book does not mention that.

2. Regarding the prove of Ito isometry for the general case (but still square integral) I can sort out the following but then stuck. Using triangular inequality,

$$|(E[\int_0^T\phi_n(t)dW(t)]^2)^{1\over2}-(E[\int_0^TfdW(t)]^2)^{1\over2}|\leq(E[\int_0^T(\phi_n(t)-f)dW(t)]^2)^{1\over2}\to0$$ we then have$$E[\int_0^TfdW(t)]^2)=\lim_{n\to\infty}E[\int_0^T\phi_n^2(t)dt],$$ but then how to get $\displaystyle\lim\limits_{n\to\infty}E[\int_0^T\phi_n^2(t)dt]=E[\int_0^Tf^2(t)dt]$?

1. Prove martingale property is preserved under $L^2$ limits. Thanks.

Answer 1

1. Any (progressively measurable) function $f$ such that $$\|f\|_{L^2(\lambda_T \otimes \mathbb{P})}^2 := \mathbb{E} \left( \int_0^T f^2(t) \, dt \right) < \infty \tag{1}$$ can be approximated in $L^2(\lambda_T \otimes \mathbb{P})$ by simple functions. This is not obvious! The convergence $$\mathbb{E} \left( \int_0^T |\phi_n(t)-f(t)| \, dt \right) \to 0 \quad (n \to \infty)$$ follows from the fact that $L^2$-convergence implies $L^1$-convergence in finite measure spaces (even if I fail to see why you call it "ideal").
2. Using the notation from $(1)$, we have by triangle inequality $$| \|\phi_n\|_{L^2(\lambda_T \otimes \mathbb{P})}-\|f\|_{L^2(\lambda_T \otimes \mathbb{P})}| \leq \|\phi_n-f\|_{L^2(\lambda_T \otimes \mathbb{P})}$$ Since $(\phi_n)_n$ is an approximating sequence, this implies $$ \|\phi_n\|_{L^2(\lambda_T \otimes \mathbb{P})}-\|f\|_{L^2(\lambda_T \otimes \mathbb{P})} \to 0 \quad (n \to \infty)$$ i.e. $$\lim_{n \to \infty} \mathbb{E} \left( \int_0^T \phi_n(t)^2 \, dt \right) = \mathbb{E} \left( \int_0^T f^2(t) \, dt \right)$$
3. You already asked this question before and since you received an answer, it would be more convenient, to ask your remaining questions about this topic there.