A problem related to Lebesgue integration.

Question

I have following two problems: Suppose $$\int_E f \, dx = 0 $$ where $ f: R \to R$ is a measurable function that is strictly positive. Show that $E$ must be a null set. Next Suppose that $E$ is a measurable subset of $[0, 1]$ for which $$\int_E x^n \sin(2x+8) \, dx = 0 $$ for each $ n= 0, 1,2,3,\ldots$. Then Show that measure of $E$ equal to zero that is $m(E)=0$. In the first problem I am thinking of use by contradiction assume that $m(E)$ not equal to zero and since $f$ is strictly positive so the integral of $f$ over $E$ is not zero which is the contradiction.Am I right? for the second problem I have no any nice idea. Do you have some ideas?

Answer 1

For the first part, let $A_n= [f>1/n]$ then $\int_{E} f \, {\rm d}x \geq \frac{1}{n}|A_n\cap E|$ therefore $|A_n\cap E|=0$ but since $\cup A_n=[0,1]$ we conclude that $|E|=0$.

For the second part, notice that your condition implies that $\int _{E} p(x)\sin(2x+8) \, {\rm d}x =0$ for any polynomial. Now, choose $p_{\epsilon}(x)$ just that $|p_\epsilon(x) -\sin(2x +8)| <\epsilon$ for all $x$.

But this implies that $$\int_E\sin^2(2x +8) \, {\rm d}x < \epsilon.$$ Which in turn gives that $\int_E\sin^2(2x +8) \, {\rm d}x = 0$ now you can use the first part.