Continuity theorem in Itô integral explanation

Question

What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text.

Answer 1

Since $g(\cdot, \omega)$ is continuous (and hence uniformly continuous on $[S,T]$), you can check that $\phi_n(\cdot, \omega) \to g(\cdot, \omega)$ uniformly on $[S,T]$. Therefore $(g(\cdot, \omega) - \phi_n(\cdot, \omega))^2 \to 0$ uniformly and this gives you the convergence of the integral.

Continuity is not strictly necessary here; piecewise continuity would be enough, or something like Riemann integrability. But you need something; if $g$ were something like the indicator of the rationals, and you unluckily chose your partition points $t_j$ to all be rational, then you would get $\phi_n = 1$ while $g=0$ almost everywhere, and the integral will not converge to 0.