I am following Folland's Real Analysis text to learn measure theory and have so far been able to understand how Borel measures are constructed over $\mathbb{R}$ from increasing, right continuous functions. That is, if we have an increasing, right continuous function $F$ then there is a unique Borel measure $\mu_F$ such that $$\mu_F((a,b]) = F(b)-F(a)$$ for any interval of the form $(a,b]$. This is all good and well, but I am left wondering what happens if we have a more abstract set that is not in a nice "interval" form? How are such sets handled? Do we simply reason them in terms of their interior?
2026-03-29 11:44:26.1774784666
How is the measure of an arbitrary set calculated?
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