 Measure theory

Main motivation: Lebesgue integral.

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The key idea, is that we can't define a measure for the power set of R. Rather, we must select a large measurable subset, and the Borel sigma algebra is a good choice that matches intuitions.

Measure theory is a branch of mathematics that deals with the systematic way of assigning a numerical "size" or "measure" to subsets of a given space. It provides a foundational framework for many areas of mathematics, particularly in integration, probability theory, and functional analysis.