Suppose we have a continuous function $f:S^2\rightarrow \mathbb{R}$ on the standard sphere. What value is given for the following integral $$\int_{\lbrace p\rbrace }f(x) \text{d$vol_{\lbrace p\rbrace}$(x)},$$
where $p\in S^2$ is any fixed point and d$vol_{\lbrace p\rbrace }$(x) is the "Riemannian volume element"? I am not sure what exactly the definition of the Riemannian volume element is in the above context, since the manifold consists of one point only. Is it just $\int_{\lbrace p\rbrace }f(x) \text{dvol(x)}:=f(p)$?
Edit: The Riemannian volume element d$vol_{\lbrace p\rbrace }$(x) is most likely meant as the Riemannian volume element of the Riemannian manifold $\lbrace p \rbrace$. (See my comment below).
Best wishes
As far as I know,
$\int_{ \lbrace P \rbrace } f(x)dvol_{ \lbrace P \rbrace }(x) = 0$
But for $\delta(x)$ to be the impulse function,
$\int_{ \lbrace P \rbrace } f(x)\delta(x)dvol_{ \lbrace P \rbrace }(x) = f(P)$