Which convergence theorem? – the monotone, dominated, and bounded convergence theorems and Fatou’s lemma, compared by hypothesis.
Constructing the integral – simple functions, nonnegative measurable functions, and integrable functions.
The convergence theorems – statements and proofs, with Egorov’s and Lusin’s theorems.
$L^1$ – basic properties of integrable functions and the modes of convergence.
Integrals over products are treated in Fubini and Tonelli, and the spaces \(L^p\) in $L^p$ spaces.