Let $\pi$ be a probability measure defined on $(\mathcal{M}, \mathcal{B}(\mathcal{M}))$, where $\mathcal{M}$ is a smooth manifold and $\mathcal{B}(\mathcal{M})$ is the Borel sigma-algebra on it. Let $\mathcal{C} \supset \mathcal{M}$ be a set.
- Can I always extend $\pi$ onto $(\mathcal{C}, \mathcal{B}(\mathcal{C}))$?
- If not, what conditions on the set $\mathcal{C}$ do I need to be able to extend $\pi$?
As clarified in your comment, your manifold is the level-set of a smooth hence continuous hence measurable function on $\mathbb{R}^n$. So the manifold is a Borel subset of $\mathbb{R}^n$. You can extend $\pi$ to all $\pi^*$ on all of $\mathbb{R}^n$ by letting $\pi^*(A)=\pi(A\cap\mathcal{M})$. If $\mathcal{C}$ itself is measurable, the restriction of $\pi^*$ to $\mathcal{B}(\mathcal{C})$ does the job. If $C$ is not measurable, you can take the measure to be $\pi^{**}$ with $\pi^{**}(A)$ being the $\pi^*$-outer measure of $A\cap C$. That this still defines a suitable measure is shown, for example, in 214A of the second volume of Fremlin's treatise.
