Let $(\Omega,\mathcal F, \mathbb P)$ be a probability space and let $\mathcal J \subseteq \mathcal F$ be a sub-$\sigma$-algebra. Show $\mathcal J $ is $\mathbb P$-trivial iff for all $X \in L^1(\mathbb P) $， $E[X \mid \mathcal J]$ is $\mathbb P$-almost surely constant.
P.S. A $\sigma$-algebra $\mathcal J \subseteq \mathcal F \,$ is $ \mathbb P$ trivial, if $\mathbb P(A) \in \{0,1\}$ for all $A \in \mathcal J$.
I am not able to get started. Any help is much appreciated.
"$\Rightarrow$" Recall that $Y:=\mathbb{E}(X \mid \mathcal{J})$ is $\mathcal{J}$-measurable and therefore $$\{Y \leq y\} \in \mathcal{J} \qquad \text{for all $y \in \mathbb{R}$}.$$ Since $\mathcal{J}$ is trivial this implies $\mathbb{P}(Y \leq y) \in \{0,1\}$ for all $y \in \mathbb{R}$. This means that $Y$ is almost surely constant.
"$\Leftarrow$": For $X := 1_I$, $I \in \mathcal{J}$, we find that $$\mathbb{E}(1_I \mid \mathcal{J}) = 1_I$$ is constant almost surely, i.e. $$\mathbb{P}(I) = \mathbb{P}(1_I =1) \in \{0,1\}.$$