Consider the following theorem and its proof.
I do not understand the claim that for each $n \in \mathbb{N}$ there exists a a measure $\nu_n$ such that $\mu_n = \mu_{n-1} + \nu_n$. Could someone please explain.
2026-05-04 08:55:10.1777884910
Need to understand why measures converging to a set function gives a measure
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Define $\nu_n(A) = \mu_n(A)-\mu_{n-1}(A)$ for all $A \in \mathcal S$. For general measures, this difference would perhaps be a signed measure. But from the assumption $\mu_n \uparrow$ it is nonnegative.
There may be a question about $\infty - \infty$. It doesn't matter, say take it to be $\infty$.