Limit of functions measurable with respect to sub-sigma algebra

Question

Let $\{f_{\alpha}\}_{\alpha}$ be a net of functions in a measure space $(X,\mathcal F,\mu)$. Suppose that each $f_{\alpha}$ is measurable with respect to a sub $\sigma$-algebra $\mathcal G$, and that the net has a limit, $f$, in the weak-* topology of $L^{\infty}(X,\mathcal F,\mu)$: that is, there is a $\mathcal F$-measuable function $f$ so that $$\int f_{\alpha} \phi d\mu \to \int f \phi d\mu$$ for every $\phi\in L^1(X,\mathcal F,\mu)$). Is $f$ $\mathcal G$-measurable?