I am reading about martingale representation theorems in J.M Steeles "Stochastic Calculus and Financial Applications" and got a bit stuck on something that is supposedly trivial, but I fail to understand.
Anyway, in the proof of his proposition 12.5 it is claimed that given a bounded variable $X \in \mathcal{F}_T$ can represented as a stochastic integral
$$
\int_0^T \phi(\omega, s) \, dB_s,
$$
where $\phi \in \mathcal{H}^2$.
Here, $\{ \mathcal{F}_t, 0 \leq t \leq T\}$ is the canonical filtration for the Brownian motion and
$
\mathcal{H}^2 =
$ All measurable adapted functions on $\mathcal{F}_T \times \mathcal{B}$ satisfying
$$
E\left[ \int_0^T f^2(\omega, t) \, dt \right] < \infty.
$$
($\mathcal{B}$ is the smallest $\sigma$-algebra containing the Borel sets on $[0,T].$)
Earlier in the book it has been proven (Dudley's representation theorem) that for any $X \in \mathcal{F}_T$ there is a $\phi \in L^2_{LOC}[0,T]$ such that $$ X = \int_0^T \phi(\omega, s) \, dB_s, $$ where $L^2_{LOC}[0,T]$ is the space of all measurable adapted functions on $\mathcal{F}_T \times \mathcal{B}$ satisfying $$ P\left[\omega : \int_0^T f^2(\omega, t) \, dt < \infty \right] = 1. $$ So given a bounded $X$ we always have the representation required to represent it using a $\phi \in L^2_{LOC}$, but how do we jump to the conclusion that $\phi \in \mathcal{H}^2$?
First I thought of the Ito isometry, but this is only applicable when we already know that $\phi \in \mathcal{H}^2$. It is not enough for the expected value of the squared integral to be finite.