Entries of moment matrices are always nonnegative?

Question

I am reading the following paper https://arxiv.org/pdf/1103.0486.pdf .

Please see p.4, the part under Theorem 2.2. (just read from the bottom of p.3 to here).

To my understanding, if the measure is a probability measure (so nonnegative measure) and the support of the measure $K$ is compact (satisfying Assumption 2.1. in that paper.), then each entry of a moment matrix is nonnegative (see p.3 bottom, the paper mentions moment matrix).

My question is if this is correct, then how to prove it? Could anyone please give me a reference?

Thanks!

Answer 1

No, a simple example is to take $\mu=\delta_{-1}$ Dirac measure on $K=\{-1\}$, (here you can take $g(t)=-(t+1)^2$), and with $u(t)= g(t)$, then entries of the moment matrix are the moment of this measure, but : $$ m_n=\int_K t^n d\mu(t)=(-1)^n. $$