Editorially prepared introduction for this redesigned journal. This is new explanatory content, separate from Pablo's original notes.
1. Measurable sets
A sigma-algebra allows countable set operations while keeping the resulting sets measurable.
Definition. A collection \(\mathcal F\subseteq\mathcal P(X)\) is a sigma-algebra if \(X\in\mathcal F\), complements of its members belong to \(\mathcal F\), and countable unions of its members belong to \(\mathcal F\).
2. Measures
Definition. A measure \(\mu:\mathcal F\to[0,\infty]\) satisfies \(\mu(\varnothing)=0\) and countable additivity on pairwise disjoint measurable sets.
3. Integration
Lebesgue integration builds the integral from measurable simple functions and extends it to more general functions through limits.
Monotone convergence theorem. If \(0\le f_n\uparrow f\) pointwise and each \(f_n\) is measurable, then \(\int f_n\,d\mu\uparrow\int f\,d\mu\).
4. The spaces Lᵖ
For \(1\le p<\infty\), \(L^p\) consists of measurable functions with \(\int |f|^p\,d\mu<\infty\), modulo equality almost everywhere.