Let $T_X = (X, \mathcal{T}_X)$ and $T_Y = (Y, \mathcal{T}_Y)$ each be a D1106: Topological space.
Let $B_X = (X, \mathcal{B}_X)$ and $B_Y = (Y, \mathcal{B}_Y)$ each be the D1838: Borel measurable space w.r.t. $T_X$ and $T_Y$, respectively.
Let $f : X \to Y$ be a D55: Continuous map with respect to $T_X$ and $T_Y$.

Let $f$ be continuous with respect to $T_X$ and $T_Y$. By definition, $\mathcal{T}_Y$ generates $\mathcal{B}_Y$. Similarly, since $\mathcal{T}_X$ generates $\mathcal{B}_X$, it follows that $\mathcal{T}_X$ is contained in $\mathcal{B}_X$. Since $f$ is continuous, result R324: Continuity characterised by preimages of open sets states that the preimages $f^{-1}(U)$ are in $\mathcal{T}_X$ and therefore in $\mathcal{B}_X$ for each $U$ in $\mathcal{T}_Y$. Since $f$-preimages of generator sets are measurable, result R1179: Measurable map iff preimages of generators measurable establishes that $f$ is measurable with respect to $B_X$ and $B_Y$. $\square$