Two conditional expectations equal almost everywhere

Question

Suppose $X$ is a continuous random variable. If $\mathbb{E}[X\,|\,\mathcal{F}]=\mathbb{E}[X\,|\,\mathcal{G}]$ almost everywhere for two sub-sigma algebra $\mathcal{F}$ and $\mathcal{G}$, does this imply $\mathcal{F}$ and $\mathcal{G}$ are set theoretically identical?

Answer 1

No. If $X,Y,Z$ are independent, $\mathcal F=\sigma (Y)$ and $\mathcal G =\sigma (Z)$ then $\mathbb E( X|\mathcal F)=\mathbb E(X|\mathcal G)=\mathbb EX$.