I am trying to prove the Stochastic Fubini's theorem which is an exercise of An Introduction to Stochastic Calculus Applied to Finance. Let $(W_t)_{t\in[0,T]}$ be a Brownian motion and $H(t,s)$ has the following properties: for each $\omega$, $H(t,s)(\omega)$ is continuous on $[0,T]^2$, and for any $s\in[0,T]$, the process $(H(t,s))_{t\in[0,T]}$ is adapted. I want to show that $\int_0^T(\int_0^TH(t,s)dW_t)ds=\int_0^T(\int_0^TH(t,s)ds)dW_t$.
As in the exercise, I showed that for any partition of $[0,T]$, by linearity
$\int_0^T(\sum_{i=0}^{N-1}H(t_i,s)(W_{t_{i+1}}-W_{t_i}))ds=\sum_{i=0}^{N-1}(\int_0^TH(t_i,s)ds)(W_{t_{i+1}}-W_{t_i})$.
Now I have to justify the limit under the integral sign. How can I justify this?