Measure Theory Problem: showing f=0 a.e.

[michael.wes]

Homework Statement

Suppose $$f\in L^2[0,1]$$ and $$\int_0^1f(x)x^n=0$$ for every n=0,1,2... Show that f = 0 almost everywhere.

Homework Equations

My friend hinted that he used the fact that continuous functions are dense in L^2[0,1], but I'm still stuck.

The Attempt at a Solution

I need help to get started.. I have tried various things with Holder's inequality but I am not getting anywhere. Thanks!

 

[Dick]

Your friends suggestion is pretty good. Continuous functions can be approximated by polynomials. What polynomials p(x) have that property?

 

[michael.wes]

Ok, I think I know how to do it now. But there is a follow-up question, which seems a lot harder, but "looks" similar:

Suppose $$f\in L^1(\mathbb{T})$$ and $$\int_0^{2\pi}f(x)x^n=0$$ for all n=0,1,2,... Show that f is 0 almost everywhere.

The hint is again to use the density of continuous functions, but also to use part of the proof of Lusin's theorem. I know this is pretty specific so I don't know if you can help... but I would appreciate it. Thanks