Weak convergence of measures

Abstract

We shall now prove Vitali’s Theorem 3.10 and Theorem 3,11. As we noted in the remark after the statement of Theorem 3.11, Vi tali’s results give non-trivial necessary and sufficient conditions in order that

$$\mathop {\lim }\limits_n \int\limits_x {|{f_n} - f|d\mu = 0} $$

((6.1))

, where (X, 𝒜, µ) is a measure space and $\left\{ {{f_n}:n = 1, \ldots } \right\} \subseteq L_\mu ^1\left( X \right)$.