I came across this problem recently.
Let $X$ be a non-negative random variable on $(\Omega,\mathscr{F},\mathbf{P})$ and let $\mathscr{G} \subseteq \mathscr{F}$ be a sub-sigma-algebra.
1) Show that $X>0 \implies \mathrm{E}[X|\mathscr{G}]>0$ almost surely.
2) Show that $\{\mathrm{E}[X|\mathscr{G}]>0\}$ is the smallest $\mathscr{G}$ measurable event that contains the event $\{{X>0}\}$ almost surely.
Now 1) can be proved defining $A=\{Y:\mathrm{E}[X|\mathscr{G}]\le0\}$. Then by the tower property $\mathrm{E}[X\mathbb{1}_A(Y)]=\mathrm{E}[\mathrm{E}(X|Y)\mathbb{1}_A(Y)]$. So $\mathrm{E}(\mathbb{1}_A(Y))=0$ a.s. so $\mathrm{E}(X|Y))>0$ a.s.
Could anyone show how to get the second part?