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For the real-valued variables here, take a $\mathcal{G}$-measurable version of $X=\mathbb{E}[Y\mid \mathcal{G}]$. Since the variables are bounded, the conditional expectation defining identity extends to the bounded $\mathcal{G}$-measurable multiplier $X$, givingThe equality case for conditional second moments now givesA nonnegative random variable with zero expectation vanishes with probability one. Thus $X=Y$ as an almost sure equality. The same proof works for square-integrable variables; boundedness is more than is needed.