I am taking a statistics course right now and we're using Hogg, McKean and Craig's Introduction to Mathematical Statistics (2013).
How do you prove Thm 1.9.1:
Let $X$ and $Y$ be random variables with moment generating functions $M_X$ and $M_Y$, respectively, existing in open intervals about $0$. Then $F_X(z)=F_Y(z)$ for all $z\in\mathbb{R}$ if and only if $M_X(t)=M_Y(t)$ for all $t\in(-h,h)$ for some $h>0$.
The book says to look at Chung (1974) - A Course in Probability Theory, but I have browsed through the book and I couldn't find the statement or the proof (I looked at Chapter 6, which deals with characteristic functions). I have looked online but could not find the proof. My statistics professor refuses to talk about the proof, even in office hours, saying it is "outside the scope of this course". I have taken upper-level undergraduate courses in real analysis, complex analysis and probability. I learned about Laplace transforms in lower-level differential equations.
The moment generating function is defined by $$M_X(t) := \int \exp(Xt) \, \mathrm{d} \mathbb{P}$$ for $t \in \mathbb{R}$, assuming $\exp(Xt) \in L^1$. This definiton makes also sense for all $t \in \mathbb{C}$ and defines a holomorphic function. Thus, your statement is a direct consequence of the identity theorem for holomorphic functions.