expectation of functions of random variable

Question

in genreral

can we claim that, for random variable' lets say discrete, X

E[f(X)]=E[g(X)] means f(x)=g(x) ?

(for all x such that p(X=x) is not zero)

Answer 1

No. Counterexample:

Let $\Omega = \{ 0, 1 \}$, $P(0) = P(1) = \frac{1}{2}$ - a symmetric coin.

f(x) = x, g(x) = 1-x.