Let $f :\mathbb{R}^{N}\to\mathbb{R}$ be a smooth function and $B_{R}$ be the $N$-dimensional ball. I consider \begin{align*}\tag{1.1}\int_{\partial B_{R}}f(x)dS\end{align*} as the surface integral of $f$ on the surface of $B_{R}$. However, I can also define \begin{align*} \chi_{\partial_{B_{R}}}:= \begin{cases} 1 \quad&\text{ if }x\in\partial B_{R}\\ 0 \quad&\text{ if }x\not\in\partial B_{R} \end{cases} \end{align*} Then, I consider \begin{align*}\tag{1.2}\int_{\mathbb{R}^{N}}f(x)\chi_{\partial B_{R}}dx \end{align*} So, my question is to consider whether (1.1) and (1.2) are the same integral or not. Specifically, I would like to know also whether $\int_{\mathbb{R}^{N}}f(x)\chi_{\partial B_{R}}dx$ and $\int_{\mathbb{R}^{N}}f(x)\chi_{\partial B_{R}}dS$ can be considered the same integral or not in this case.
Any help is greatly appreciated! Thank you
They are not. For the surface $\partial B_{R}$, it is of measure zero by taking the Lebesgue measure, so \begin{align*} \int_{\mathbb{R}^{n}}f(x)\chi_{\partial B_{R}}(x)dx=0. \end{align*}