Measure of diagonal in measure preserving dynamical system

Question

Suppose that we have dynamical system $(X,T,B,\mu)$, where $B$ is Borel set and $\mu$ propability invariant measure. Is it true, that $\mu \times \mu(\Delta) > 0 $, where $\Delta = \{(x,x) | \: x \in X \}$ and $\mu \times \mu (A_0\times A_1) := \mu(A_0)\mu(A_1)$ is equivalent to $\mu$ being dirac measure i.e. $\mu(q) = 1$ for some $q \in X$?