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I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for strong or total variation convergence. I have two questions.

1. Are there some equivalent formulations for those notions of convergence too? In particular, is it true that $\mu_n$ strongly converges to $\mu$ iff $E_{\mu_n} f\to E_\mu f$ for every bounded measurable function $f$? According to an answer to this question, the left-to-right direction holds if strong convergence is strengthened to total variation convergence.

2. Are there some nice characterizations of these topologies that are not stated in terms of convergent sequences?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for strong or total variation convergence. I have two questions.

1. Are there some equivalent formulations for those notions of convergence too? In particular, is it true that $\mu_n$ strongly converges to $\mu$ iff $E_{\mu_n} f\to E_\mu f$ for every bounded measurable function $f$? According to an answer to this question, the left-to-right direction holds if strong convergence is strengthened to total variation convergence.

2. Are there some nice characterizations of these topologies that are not stated in terms of convergent sequences?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for strong or total variation convergence. I have two questions.

1. Are there some equivalent formulations for those notions of convergence too? In particular, is it true that $\mu_n$ strongly converges to $\mu$ iff $E_{\mu_n} f\to E_\mu f$ for every bounded measurable function $f$? According to an answer to this question, the left-to-right direction holds if strong convergence is strengthened by total variation convergence.

2. Are there some nice characterizations of these topologies that are not stated in terms of convergent sequences?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for strong or total variation convergence. I have two questions.

1. Are there some equivalent formulations for those notions of convergence too? In particular, is it true that $\mu_n$ strongly converges to $\mu$ iff $E_{\mu_n} f\to E_\mu f$ for every bounded measurable function $f$? According to an answer to this question, the left-to-right direction holds if strong convergence is strengthened to total variation convergence.

2. Are there some nice characterizations of these topologies that are not stated in terms of convergent sequences?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for strong or total variation convergence. I have two questions.

1. Are there some equivalent formulations for those notions of convergence too? In particular, is it true that $\mu_n$ strongly converges to $\mu$ iff $E_{\mu_n} f\to E_\mu f$ for every bounded measurable function $f$? According to an answer to this question, the left-to-right direction holds if strong convergence is strengthened by total variation convergence.

2. Are there some nice characterizations of these topologies that are not stated in terms of convergent sequences?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.

The Wikipedia article on convergence of measures defines three kinds of convergence: total variation, strong, and weak. For weak convergence, a number of equivalent formulations are given, but not for strong or total variation convergence. I have two questions.

1. Are there some equivalent formulations for those notions of convergence too? In particular, is it true that $\mu_n$ strongly converges to $\mu$ iff $E_{\mu_n} f\to E_\mu f$ for every bounded measurable function $f$? According to an answer to this question, the left-to-right direction holds if strong convergence is strengthened by total variation convergence.

2. Are there some nice characterizations of these topologies that are not stated in terms of convergent sequences?