[proofplan] We prove the contrapositive estimate directly from countable additivity. If $W$ is wandering, then its backward iterates $T^{-n}(W)$ are pairwise disjoint measurable subsets of $X$. Measure preservation makes every one of these sets have measure $\mu(W)$, so finite total measure of $X$ forces the infinite sum $\sum_{n=0}^{\infty}\mu(W)$ to be finite. This is possible only when $\mu(W)=0$. [/proofplan] [step:Form the disjoint backward orbit of the wandering set] Let $W \in \mathcal{B}$ be a wandering set. For each integer $n \geq 0$, define \begin{align*} W_n := T^{-n}(W) \in \mathcal{B}. \end{align*} By the definition of a wandering set, the sets $(W_n)_{n=0}^{\infty}$ are pairwise disjoint measurable subsets of $X$. [/step] [step:Use measure preservation to give every backward iterate the same measure] Since $T$ is measure-preserving, for every set $A \in \mathcal{B}$ one has $\mu(T^{-1}(A))=\mu(A)$. Applying this identity inductively to $A=W$ gives, for every integer $n \geq 0$, \begin{align*} \mu(W_n)=\mu(T^{-n}(W))=\mu(W). \end{align*} [/step] [step:Apply countable additivity inside the finite measure space] Define the measurable set \begin{align*} Y := \bigcup_{n=0}^{\infty} W_n \subseteq X. \end{align*} Because the sets $(W_n)_{n=0}^{\infty}$ are pairwise disjoint, countable additivity of the measure $\mu$ gives \begin{align*} \mu(Y) = \sum_{n=0}^{\infty} \mu(W_n) = \sum_{n=0}^{\infty} \mu(W). \end{align*} Since $Y \subseteq X$, monotonicity of $\mu$ and the hypothesis $\mu(X)<\infty$ give \begin{align*} \sum_{n=0}^{\infty} \mu(W) = \mu(Y) \leq \mu(X) < \infty. \end{align*} [/step] [step:Conclude that the wandering set has zero measure] The number $\mu(W)$ is non-negative because $\mu$ is a measure. If $\mu(W)>0$, then the partial sums satisfy \begin{align*} \sum_{n=0}^{N} \mu(W) = (N+1)\mu(W) \end{align*} for every integer $N \geq 0$, and these partial sums tend to $+\infty$ as $N \to \infty$. This contradicts the finiteness of $\sum_{n=0}^{\infty}\mu(W)$. Therefore $\mu(W)=0$, so every wandering set has $\mu$-measure zero. [/step]