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?