Lebesgue's theorem

Lebesgue's theorem states that if a function is integrable on a certain interval (meaning its total area under the curve is finite), then it can be approximated as closely as desired by a sequence of simpler functions called step functions. These step functions are constant over small regions and are easier to work with. Essentially, the theorem assures us that complex functions can be "built up" from simple pieces, allowing for more effective analysis and integration. This foundation underpins many advanced topics in mathematical analysis and ensures that Lebesgue integration aligns well with our intuitive understanding of area.