This is a note for Rudin’s real and complex analysis chapter 1. The key concepts are \(\sigma\)-algebra, measure (\(\mu\)) zero, and linear combination. The three concepts bring me abstract integration. The \(\sigma\)-algebra makes that countable sum and measure of complement (subtract measure) can be possible. Measure zero completes the system. linear combination integrates a measurable function.
After a measure space established, Lebesgue’s monotone convergence theorem, Fatou’s lemma, and Lebesgue’s dominant convergence theorem follow. Although three theorems do not contain integral in their name, they insist that pointwise convergent sequence of functions is also converging their integral of the limit of functions. Lebesgue’s monotone convergence theorem requires a monotonous increment of series of functions and Lebesgue’s dominant convergence theorem requires upper bound function. Fatou’s lemma is the inequality of the lower limit.
Lebesgue’s monotone convergence theorem can be proved by the fact all \(L_1(\mu)\) has convergent simple function sequence. \(f_n\) > simple functions holds inequality of \(\lim\int f_n \geq \lim\int S_n = \int f\). Fatous’s lemma \(\lim \inf \int \lvert fn - f \rvert \leq\int \lim \inf \lvert f_n - f \rvert\) is used for proof of Lebesgue’s dominant convergence.
