Lebesgue's theorem on the integral

Lebesgue's theorem states that if you have a sequence of functions that get closer and closer to a specific function, and all these functions are bounded and measurable, then the integral (area under the curve) of their limit function is the same as the limit of their individual integrals. In simple terms, under certain conditions, you can interchange the process of taking a limit and calculating the integral—doing one before the other gives the same result. This theorem is fundamental in analysis because it ensures consistent integration for sequences of functions.