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

For $B\in\mathcal{F}$ put $\mu(B)=\sup_{j}\lambda_{j}(B)$. This defines a function $\mu$ on $\mathcal{F}$ (by Predicative Formulas for Suprema and Infima, and Maps Defined by Suprema and Infima (iii) and Maps and Relations Given by Formulas (i); here and below the defining expressions are predicative as formula trees by The Graph Reading of a Formula Tree Is Predicative and Agrees with Its Reading, So the Reading Is Predicative (i) and The Graph Reading of a Formula Tree Is Predicative and Agrees with Its Reading, So the Reading Is Predicative (ii), and $1$ and $j+1$ lie in $\mathbb{N}$ as successors), and its values lie in $[0,\infty]$, since $\mu(B)\ge\lambda_{1}(B)\ge0$. As $\lambda_{j}(\emptyset)=0$ for every $j$ by Measure, Measure Space, and Probability Measure (i), $\mu(\emptyset)=0$.

Let $(B_{m})$ be a pairwise disjoint sequence in $\mathcal{F}$, and let $w:\mathbb{N}\times\mathbb{N}\to[0,\infty]$ be the function $w(j,m)=\lambda_{j}(B_{m})$ (a function by Maps and Relations Given by Formulas (ii)). By hypothesis $w(j,m)\le w(j+1,m)$ for all $j,m\in\mathbb{N}$. Countable additivity of each $\lambda_{j}$ (Measure, Measure Space, and Probability Measure (i)) and Sums of Sequences in [0,∞]: Real Series, Linearity, Comparison, Rearrangement, Double Series and Monotone Convergence (xiv), applied to $w$, give

$$\mu\Bigl(\bigcup_{m\in\mathbb{N}}B_{m}\Bigr)=\sup_{j}\lambda_{j}\Bigl(\bigcup_{m\in\mathbb{N}}B_{m}\Bigr)=\sup_{j}\sum_{m\in\mathbb{N}}\lambda_{j}(B_{m})=\sum_{m\in\mathbb{N}}\sup_{j}\lambda_{j}(B_{m})=\sum_{m\in\mathbb{N}}\mu(B_{m}).$$

So $\mu$ is a measure on $(X,\mathcal{F})$ by Measure, Measure Space, and Probability Measure (i), and it satisfies the displayed equation of (i) by construction.