If $f$ is a function in $L^2([0,1]^m)$, W is one-dimensional Brownian motion, $a,b \in [0,1]$, are the following two integrals equal?
$$\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2} \left(\int_a^bf(t_1, t_2, \cdots, t_{m-1}, s)ds\right) dW_{t_1}dW_{t_2}\cdots dW_{t_{m-1}}$$
$$\int_a^b\left(\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2}f(t_1, t_2, \cdots, t_{m-1}, s)dW_{t_1}dW_{t_2}\cdots dW_{t_{m-1}}\right)ds$$
I know we can use Fubini's theorem to interchage integral order in deterministic cases, but since here we have mixed deterministic and stochastic integral, I don't know how to make justification.
Thank you for help.