The Setwise Supremum of an Increasing Sequence of Measures Is a Measure

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ be a measurable space, and let $(\lambda_{j})_{j\in\mathbb{N}}$ be a sequence of measures on $(X,\mathcal{F})$ that increases setwise:

$$\lambda_{j}(B)\le\lambda_{j+1}(B)\qquad\text{for all }j\in\mathbb{N}\text{ and }B\in\mathcal{F}.$$

(i) The setwise supremum is a measure. There is a measure $\mu$ on $(X,\mathcal{F})$ such that

$$\mu(B)=\sup_{j\in\mathbb{N}}\lambda_{j}(B)\qquad\text{for every }B\in\mathcal{F}.$$