Practice problem

Measure and Integration Theory

Let $X = (X,\Sigma,\mu)$ be a measure space. Let $\{B_n:n\geq1\}$ be a countable sequence of measurable sets of $X$ .

Which of the following two statements is always true?

I. If $\{B_n:n\geq1\}$ increases to (its union) $B$ , then $\lim_{n\to\infty} \mu(B_n) = \mu(B)$ .
II. If $\{B_n:n\geq1\}$ decreases to (its intersection) $B$ , then $\lim_{n\to\infty} \mu(B_n) = \mu(B)$ .