= Solution
For a measurable map, each preimage $\theta^{-1}(A)$ lies in $\mathcal E$, so $\mu_\theta$ is well-defined and nonnegative. The preimage of the <empty set> is empty, and preimages commute with disjoint countable <unions>, giving
$$
\mu_\theta\!\left(\bigcup_{j=1}^\infty A_j\right)=\mu\!\left(\bigcup_{j=1}^\infty\theta^{-1}(A_j)\right)=\sum_{j=1}^\infty\mu_\theta(A_j)
$$
for pairwise disjoint measurable $A_j$. Thus \b[$\mu_\theta$ is a <measure>], the <pushforward measure> $\theta_*\mu$. Its total mass is $\mu(E)$, because $\theta^{-1}(E)=E$. Finiteness is not needed for this construction, but is used in the uniqueness argument next.
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