approximating a measure by smooth measures

Question

Let $M\subset \mathbb R^n$ be a compact $C^\infty$ manifold with boundary $S\in C^\infty$ with a surface measure $ds$ induced from $\mathbb R^n$ and $\eta$ is a non negative finite Borel measure on $M$. I want to approximate $\eta$ by a sequence of smooth measures $\eta_k=g_k(x)ds$ with non negative functions $g_k\in C^\infty(M) $ in a sense that for every continuous function $f$ on $M$ $$ \lim_{k\to\infty} \int_M f(x)\,d\eta_k=\int_M f(x)\,d\eta. $$ What is a simple/standard way to do it? A straightforward idea would be to take a "convolution" of $\eta$ with some sort of delta-like sequence of smooth functions. But part of the measure can be concentrated on the boundary and I'm not sure how to deal with it. Say, $\eta$ can contain delta function $\delta(x-x_0)$ for some $x_0\in S$.