Let $f(x,y)$ be a measurable function defined on $(\mathbb{R}^2, \mathcal{B}(\mathbb{R}^2))$. Define $C_{\epsilon} = \{y:d(y, y_0)\leq \epsilon\}$, then can we say for sure that the integration $$ \int_{\mathbb{R}\times C_\epsilon} f(x,y) \mu(dx)\nu(dy) \rightarrow 0 \quad (\epsilon \rightarrow 0) $$ if we assume $\nu(dy)$ has no atoms. Or, can it be $$ \int_{\mathbb{R}\times C_\epsilon} f(x,y) \mu(dx)\nu(dy) \rightarrow \int_{\mathbb{R}} f(x, y_0) \mu(dx) \quad (\epsilon \rightarrow 0) $$ Which one is right? Thanks!
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