If $E[U|X]=0$ then $[XU] = 0$
If $E[XU]=0$ then $[U|X] = 0$
Which of the two statements above are true? This is my thought process for the first one:
if $E[U|X]=0$ then $E[U]=0$ and if U and X are independent, then $E[XU]=E[X]E[U]=E[X]*0=0$
which means the first statement is true if X and U are independent.
Is this the right way to think about it? What about the second one?