Let $(X,\mathfrak{B}_X,\mu_X)$ and $(Y,\mathfrak{B}_Y,\mu_Y)$ be $\sigma$-finite measure spaces. Then there exists a unique measure $\mu_X\times\mu_Y$ on $\mathfrak{B}_X\times\mathfrak{B}_Y$ that obeys $\mu_X\times\mu_Y(E\times F)=\mu_X(E)\times\mu_Y(F)$ whenever $E\in \mathfrak{B}_X$ and $F\in \mathfrak{B}_Y$.
That is the existence and uniqueness of product measure.
I want to ask if I remove condition"$\sigma$-finite", is the conclusion still correct? If not, please give a counterexample.