Assuming countable choice, Borel probability measures on Polish spaces are inner regular

The empty $P$ is immediate, so suppose $P\mathrm{\ne }\mathrm{\varnothing }$ and enumerate a countable dense set as $(x_m)$. For each $r\ge 1$, continuity from below chooses a finite union $U_r$ of $2^{-r-2}$-balls centered at the $x_m$ with $\mu (P\setminus U_r)<\epsilon 2^{-r-2}$. Countable choice makes these choices simultaneously.

The set $K_0={\bigcap }_r\overline{U_r}$ is closed. Countable subadditivity gives $\mu (P\setminus K_0)<\epsilon \mathrm{/}2$. For every scale, one of the finite closed-ball covers from step 1.1 covers $K_0$; choosing one point of $K_0$ from each nonempty member of that finite cover and doubling the radius gives a finite net with centres in $K_0$. Hence $K_0$ is totally bounded and [F3] makes it compact.

Every open $G$ contains a compact $K\subseteq G$ losing less than $\epsilon$: for $G=P$ use $K_0$; otherwise intersect $K_0$ with the increasing closed sets $\{x:d(x,P\setminus G)\ge 1\mathrm{/}n\}$. Their union is $K_0\cap G$, so [F2] gives one with the required loss.

Let $\mathcal{R}$ be the Borel sets which, for every $\delta >0$, have compact $K\subseteq A\subseteq G$ with $G$ open and $\mu (G\setminus K)<\delta$. Step 3.1 puts every open set in $\mathcal{R}$. The tight compact set from step 2.1 shows that complements remain in $\mathcal{R}$: from $K\subseteq A\subseteq G$, use $K_0\setminus G\subseteq P\setminus A\subseteq P\setminus K$ and bound the loss by $\mu (P\setminus K_0)+\mu (G\setminus K)$.

For pairwise disjoint $A_j\in \mathcal{R}$, continuity from below makes the measure of the tail of ${\bigcup }_j{A}_j$ arbitrarily small. A finite union of compact inner approximants handles that tail, while the union of the open outer approximants has loss bounded by the summable errors. Thus $\mathcal{R}$ is a lambda-system. Open sets are a pi-system generating the Borel sigma-algebra, so [F4] gives $\mathcal{B}(P)\subseteq \mathcal{R}$. The compact inner approximant for $A$ proves the statement.