 Given the above definition of total variation distance, a sequence μ"n" of measures defined on the same measure space is said to converge to a measure "μ" in total variation distance if for every "ε" > 0, there exists an "N" such that for all "n" > "N", one has that

For formula_13 a measurable space, a sequence μ"n" is said to converge strongly to a limit "μ" if

for every set formula_20.For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ"n" of measures on the interval [−1, 1] given by μ"n"("dx") = (1+ sin("nx"))"dx" converges strongly to Lebesgue measure, but it does not converge in total variation.In mathematics and statistics, weak convergence is one of many types of convergence relating to the convergence of measures.