# Necessity and sufficiency of martingale

Let's consider probability space $$(\Omega,\Sigma,P)$$ with sequence of random bounded variables $$(X_n)$$. We assume also that $$S_n:=X_1+X_2+...+X_n$$. We put filtration $${\displaystyle {\mathcal {F_n}}}=\Sigma.$$

I want to find necessity and sufficiency conditions that S_n is a martingale with respect to filtration $${\displaystyle {\mathcal {F_n}}}=\Sigma.$$

Sufficient Condition

$$E(S_{n+1}| {\displaystyle {\mathcal {F_n}}})=E(S_{n}| {\displaystyle {\mathcal {F_n}}})+E(X_{n+1}| {\displaystyle {\mathcal {F_n}}})=S_n+E(X_{n+1}).$$

So the Sufficient Condition would be that $$S_n$$ is $${\displaystyle {\mathcal {F_n}}}$$ measurable and for any $$n$$ :$$E(X_n)=0$$.

Is my justification correct ? What can i say about necessity condition ?

It is not true that $$E(X_{n+1}|\mathcal F_n)=EX_{n+1}$$. All your random variables are measurable w.r.t. $$\mathcal F_n$$ so $$E(X_{n+1}|\mathcal F_n)=X_{n+1}$$. Hence a necessary and sufficient condition for $$\{S_n\}$$ to be a martinagle is $$X_n=0$$ almost surely for all $$n$$.