Let $f$ be Lebesgue measurable, and $\int_a^b x^\alpha f(x) = 0$ for every $\alpha\ge 0$.
How do I show that $f(x)=0 ~ a.e.$
and if the condition change "$\alpha\ge 0$" to "$\alpha\ge k$ for some $k\ge 1$", if the statement is also true?
Let $f$ be Lebesgue measurable, and $\int_a^b x^\alpha f(x) = 0$ for every $\alpha\ge 0$.
How do I show that $f(x)=0 ~ a.e.$
and if the condition change "$\alpha\ge 0$" to "$\alpha\ge k$ for some $k\ge 1$", if the statement is also true?
For $f$ a polynomial, both cases should be true. In the first case, by the linearity of integration, we have $\int f p = 0$ for any $p \in \mathbb{R}[x]$, in particular $\int f^2 = 0$ giving that $f^2 \equiv 0$ and the result.
In the second case, we have $\int f p = 0$ for any $p$ with lowest term $x^k$. In particular we have $\int f^2 x^{2n} = 0$, for some $n \gg 0$, so same trick applies.
For general $f$, I think polynomials are dense in $L^1$, from which the result should follow, namely if $f_i \in \mathbb{R}[x]$ $\rightarrow f$, we have $0 = \int (x^{2n})f_i f \rightarrow \int (x^{2n}) f^2$.