Consider Lebesgue measure (lambda) on the space (([a, b], mathscr{B}[a, b])) and assume there on that (f

Question:

Consider Lebesgue measure $\lambda$ on the space $([a, b], \mathscr{B}[a, b])$ and assume there on that $f \in \mathcal{L}^{1}([a, b], \lambda)$ satisfies $\int_{a}^{b} x^{n} f(x) d x=0$ for all $n=0,1,2, \ldots$. Show that $\left.fight|_{[a, b]}=0$ is Lebesgue almost everywhere.

[ use Weierstraß' approximation theorem, Theorem 28.6.]

Data from theorem 28.6

Theorem 28.6 (Weierstra) Polynomials are dense in C[0, 1] w.r.t. uniform convergence. Proof (S. N. Bernstein)