Here $T$ is what, the natural Borel sigma-algebra? And the measures are probability measures? In my answer I assume these.
For this: $\mu(A)=\nu(f^{-1}(A))$, I might write $\mu = f(\nu)$ or maybe $\mu = f_*(\nu)$ and say that $\mu$ is the image of $\nu$ under $f$. Every probability measure on $2^\omega$ is an image of Lebesgue measure. Lebesgue measure is an image of a measure $\nu$ if and only if $\nu$ is atomless.