Questions about shift-invariant measures in ${\bf N}$

Question

Let $P$ be a shift invariant diffused probability measure defined on powerset of all natural numbers ${\bf N}$(see, van Douwen, Eric K. (1992). Finitely additive measures on ${\bf N}$. Topology Appl.${\bf 47(3)}$ 223--268. MR1192311, DOI: 10.1016/0166-8641(92)90032-U). Let $\mu_1=\mu_2=P$.

${\bf Question 1.}$ Is Fubini theorem valid for the product measure $\mu_1\times \mu_2$?

Answer 1

This example is due to D. Let consider a function $f(k,n): N^2 \to R$ defined by $f(k,n)=1$ for $k \le n$ and $f(k,n)=0$, otherwise. Then $\int_N(\int_Nf(k,n)dP(k))dP(n)=0$ and $\int_N(\int_Nf(k,n)dP(n))dP(k)=1$.