Could someone please point out how we can recover or construct functions, $g(X)$, that satisfy the following equality?
$$E[g(X)]=E(X)$$
Here, X is any random variable with mean $E(X)$ and $g(X)$ is any well behaved function, that we need to find that satisfies the above equation.
In words, we are looking to find functions of random variable such that the expectation of the function is equal to the expectation of the random variable.
If we need to make any further assumptions, please state them explicitly.
Please let me know if anything is not clear or if you need any further information.