Suppose that \{g_n\}_{n=1}^\infty is a sequence of positive continuous functions on I=[0,1], \mu is a positive Borel measure on I, m is the standard Lebesgue measure, and
(i) \lim_{n\rightarrow\infty} g_n =0 \qquad a.e. [m];
(ii) \int_{I}{g_n}\;dm = 1 \qquad for all n\in\mathbb{N};
(iii)...