Assume that we have a sequence of signed measures $\mu_n$ on $[0,1]$ that converge weak(-*) to $\mu$, that means:
For all continuous and bounded functions $f:[0,1] \rightarrow R$ we have $\int_0^1 f(x) \mu_n(dx) \rightarrow \int_0^1 f(x) \mu(dx)$.
Question
Does then also the product measure on $[0,1] \times [0,1]$ $\mu_n \otimes \mu_n$ weak(-*-)converge to $\mu \otimes \mu$?
i.e: for all continuous and bounded functions $f:[0,1] \times [0,1] \rightarrow R$ do we have $\int_{[0,1] \times [0,1]} f(x,y) \mu_n(dx)\otimes \mu_n(dy) \rightarrow \int_{[0,1] \times [0,1]} f(x) \mu(dx)\otimes \mu(dy)$.
Alternative Question
If this is not true for all such $f$ is it for example true for $f(x,y)=f(x-y)$? (or/and for $f$ being smooth)
Remark
This is true if $\mu_n$ and $\mu$ are probability measures. But is this also true for signed measures?
Prove it first for $f(x,y)=g(x)h(y)$ then for a finite linear combinations of such functions. Then use the fact that finite linear combinations of such functions are dense in $C([0,1]^2)$ (which follows from Weierstrass theorem).