Question on comparision of integrals

Question

I have this integral

$$\int_0^1 U(M)dx\ge \int_0^1 U(N)dx$$

for any function U.

According to this integral, can I say that $U(M)\ge U(N)$?

Is this true?

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Edit:

If real integral is as follows:

$$\int_0^1 \int_0^x g(z)M(z)dz dx\ge \int_0^1 \int_0^x h(z)N(z)dzdx$$

Then similarly $U(M)=\int_0^x g(z)M(z)dz \ge \int_0^x h(z)N(z)dz=U(N)$ Can I say this?

Answer 1

No, you cannot say this. What you obtain is that $$\int_0^1\int_0^x\left\{g(z)M(z)-h(z)N(z)\right\}\,dz\,dx\ge0$$What this tells you is that the total integral (area / region) is positive. But this doesn't guarantee that for some value of $z$, that the function dips into the negative range.