} \left| \int_X f_n(x) , d\mu - \int_X f(x) , d\mu \right| &\leq \int_X |f_n(x) - f(x)| , d\mu \ &= \int_A |f_n(x) - f(x)| , d\mu + \int_{X \setminus A} |f_n(x) - f(x)| , d\mu \ &\leq \int_A \epsilon / \mu(A) , d\mu + \int_{X \setminus A} g(x) , d\mu \ &\leq \epsilon + \int_{X \setminus A} g(x) , d\mu \ &\leq \epsilon + \epsilon \ &= 2 \epsilon. \end{align