The Dieudonne club-set function is a Borel measure

Let $\mathcal{D}$ be the sets $A\subseteq Y$ for which either $A$ or $Y\setminus A$ contains a club. Two disjoint sets cannot both contain clubs, since two clubs intersect by [L1]. Complements preserve $\mathcal{D}$. For a sequence $(A_n)\subseteq \mathcal{D}$, if some $A_n$ contains a club then so does ${\bigcup }_n{A}_n$; otherwise each complement contains a club and [L1] puts a club in the complement of the union. Thus $\mathcal{D}$ is a sigma-algebra.

Every open $U\subseteq Y$ belongs to $\mathcal{D}$: if the closed complement is unbounded, it is club; if it is bounded by $\alpha$, then the tail $[\alpha +1,{\omega }_1)$ is a club contained in $U$. Hence $\mathcal{B}(Y)\subseteq \mathcal{D}$.

On $\mathcal{D}$, the displayed $0$-$1$ rule is countably additive. Indeed, among pairwise disjoint $A_n$ at most one has value $1$; if none does, the intersection of club subsets of their complements is club by [L1], so their union has value $0$. Also $m(Y)=1$. Restriction $E\mapsto E\cap Y$ is a sigma-homomorphism from $\mathcal{B}(X)$ to $\mathcal{B}(Y)$, proving the assertion for $\bar{m}$.