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2 added 85 characters in body

If the integrals are finite, we can approximate by integrals on compacta; I'm assuming from the notation you're using that $\alpha_n, \alpha$ are functions of bounded variation, and the integrals are Riemann-Stieltjes. Then there is a standard result that says that if $\alpha_n \rightarrow \alpha$ pointwise on a compact interval and the total variation of $\alpha_n$ is uniformly bounded for all $n$, then $\int f \ d\alpha_n \rightarrow \int f \ d\alpha$.

If the uniformly bounded property doesn't hold, then the variation of $\alpha$ could be infinity and the integral need not converge.

Edit: Question has ambiguous notation, so I'm making some loose assumptions here.

1

If the integrals are finite, we can approximate by integrals on compacta; I'm assuming from the notation you're using that $\alpha_n, \alpha$ are functions of bounded variation, and the integrals are Riemann-Stieltjes. Then there is a standard result that says that if $\alpha_n \rightarrow \alpha$ pointwise on a compact interval and the total variation of $\alpha_n$ is uniformly bounded for all $n$, then $\int f \ d\alpha_n \rightarrow \int f \ d\alpha$.

If the uniformly bounded property doesn't hold, then the variation of $\alpha$ could be infinity and the integral need not converge.