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I'm struggling in finding the hypotheses for weak convergence to imply setwise convergence, for a sequence of induced measures.

Consider a standard probability triple $(\Omega,\mathcal{B},\mu)$. Let $f_n\rightarrow f$ pointwise. Let $\mu_f$ be the measure induced by f and $\mu_n$ be the measure induced by $f_n$ for each n. Finally let $\mu_n$ converge weakly to $\mu_f$, that is, $\int_{\Omega} h \,d\mu_n\rightarrow \int_\Omega h \, d\mu_f$ for all bounded continuous $h$.

Under what additional hypothesis $\mu_n$ converges setwise to $\mu_f$: $\lim_n \mu_n(A)=\mu_f(A)$ for all $A\in\mathcal{B}$?

I'm struggling in finding the hypotheses for weak convergence to imply setwise convergence, for a sequence of induced measures.

Consider a standard probability triple $(\Omega,\mathcal{B},\mu)$. Let $f_n\rightarrow f$ pointwise. Let $\mu_f$ be the measure induced by f and $\mu_n$ be the measure induced by $f_n$ for each n. Finally let $\mu_n$ converge weakly to $\mu_f$, that is, $\int_{\Omega} h \,d\mu_n\rightarrow \int_\Omega h \, d\mu_f$ for all bounded continuous $h$.

Under what additional hypothesis on $f$ and $f_n$, $\mu_n$ converges setwise to $\mu_f$: $\lim_n \mu_n(A)=\mu_f(A)$ for all $A\in\mathcal{B}$?

I'm struggling in finding the hypotheses for weak convergence to imply setwise convergence, for a sequence of induced measures.

Consider a standard probability triple $(\Omega,\mathcal{B},\mu)$. Let $f_n\rightarrow f$ pointwise. Let $\mu_f$ be the measure induced by f and $\mu_n$ be the measure induced by $f_n$ for each n. Finally let $\mu_n$ converge weakly to $\mu_f$: $\int_{\Omega} fd\mu_n\rightarrow \int_\Omega fd\mu_f$ for all bounded continuous f.

Under what additional hypothesis $\mu_n$ converges setwise to $\mu_f$: $\lim_n \mu_n(A)=\mu_f(A)$ for all $A\in\mathcal{B}$?

I'm struggling in finding the hypotheses for weak convergence to imply setwise convergence, for a sequence of induced measures.

Consider a standard probability triple $(\Omega,\mathcal{B},\mu)$. Let $f_n\rightarrow f$ pointwise. Let $\mu_f$ be the measure induced by f and $\mu_n$ be the measure induced by $f_n$ for each n. Finally let $\mu_n$ converge weakly to $\mu_f$, that is, $\int_{\Omega} h \,d\mu_n\rightarrow \int_\Omega h \, d\mu_f$ for all bounded continuous $h$.

Under what additional hypothesis $\mu_n$ converges setwise to $\mu_f$: $\lim_n \mu_n(A)=\mu_f(A)$ for all $A\in\mathcal{B}$?

I'm struggling in finding the hypotheses for weak convergence to imply setwise convergence, for a sequence of induced measures.

Consider a standard probability triple $(\Omega,\mathcal{B},\mu)$. Let $f_n\rightarrow f$ pointwise. Let $\mu_f$ be the measure induced by f and $\mu_n$ be the measure induced by $f_n$ for each n. Finally let $\mu_n\rightharpoonup \mu$ (i.e $\int_{\Omega} fd\mu_n\rightarrow \int_\Omega fd\mu$).

Under what additional hypothesis $\mu_n$ converges setwise to $\mu$: $\lim_n \mu_n(A)=\mu(A)$ for all $A\in\mathcal{B}$?

I'm struggling in finding the hypotheses for weak convergence to imply setwise convergence, for a sequence of induced measures.

Consider a standard probability triple $(\Omega,\mathcal{B},\mu)$. Let $f_n\rightarrow f$ pointwise. Let $\mu_f$ be the measure induced by f and $\mu_n$ be the measure induced by $f_n$ for each n. Finally let $\mu_n$ converge weakly to $\mu_f$: $\int_{\Omega} fd\mu_n\rightarrow \int_\Omega fd\mu_f$ for all bounded continuous f.

Under what additional hypothesis $\mu_n$ converges setwise to $\mu_f$: $\lim_n \mu_n(A)=\mu_f(A)$ for all $A\in\mathcal{B}$?