# Ito isometry property for two variable functions.

As we know about isometry property of Ito integral as in {Verifying Ito isometry for simple stochastic processes} i.e. $$\mathbb{E}\left[ \left( \int_0^T X_t~dB_t\right)^2 \right] = \mathbb{E} \left[ \int_0^T X^2_t~dt \right],$$ for any $X\in L^2.$

I am interested to know, HOW CAN we extend this property for BROWNIAN SHEET in place of Brownian motion i.e. if $f(x,t)\in L^2{([0,1], [0,T])},$ can we find isometry property as

$$\mathbb{E}\left[ \left(\int_0^1 \int_0^T f(y,s)~W(dy,ds)\right)^2 \right] = \mathbb{E} \left[\int_0^1 \int_0^T f(y,s)^2 ~dy ds \right],$$

where W(x,t) is the Brownian sheet with $$\mathbb{E}|W(x,t)|=0$$ and $$\mathbb{E}|W(x,t) W(y,s)|=|t\wedge s||x\wedge y|,$$ where $t\wedge s$= min(t,s)