If
we call
a
moment of
.
Problem: Prove that if
have the same moments (for
) then the must be the same function.
Proof: Define the functional
by
. We first prove that
is continuous.
Lemma: is continuous.
Proof: Note that if
we're done since the integral of a continuous non-negative function is zero only when the function is identically zero. Thus, we may assume that
Let
be given. Then choosing
such that
we see that
The conclusion follows.
.
But notice by our assumption that
for every
or equivalently
.
But, we in fact see that given any polynomial
that
And since by Weirstrass's theorem we have that the polynomials are dense in
we have that
agrees with the zero function on a dense subset of it's domain. It follows that
for every
. The conclusion follows from
this.
Does that look right? I feel like I made this way harder than it need be.