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.