= Measure theory
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Measure theory studies <measures>, which assign sizes to <measurable sets>, and the <Lebesgue integral>, which accumulates values weighted by those sizes. The same framework describes <Lebesgue measure> on the <real numbers>, <counting measure> on discrete collections, and <probability measures> on possible outcomes. The choice of <measure> determines what counts as large; the choice of <sigma-algebra> determines what can be measured.
A striking example separates size from proximity: the <rational numbers> in $[0,1]$ form a <dense set> with <Lebesgue measure> zero, although the whole interval has <Lebesgue measure> one. This distinction makes <almost everywhere> statements useful: exceptional <null sets> can be ignored in a <Lebesgue integral> even when they meet every open subinterval. The <monotone convergence theorem> and <dominated convergence theorem> then give precise conditions for passing a <limit> through an <integral>.
A historical starting point is Lebesgue's 1902 thesis, https://www.ime.usp.br/~glaucio/mat5798/textos/aula1.pdf#page=2[Intégrale, Longueur, Aire (original title page reproduced in USP lecture notes)]{ref}. Its title connects <integration> with the geometric task of assigning sizes.
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