Lebesgue's Dominated Convergence Theorem

Lebesgue’s Dominated Convergence Theorem states that if a sequence of functions converges point-by-point to a limit function and all these functions are bounded by some integrable (finite integral) function, then the integral (area under the curve) of the limit function equals the limit of the integrals of the functions. In essence, under these conditions, you can interchange the limit and integral operations without changing the result, which simplifies calculations and ensures consistency in analysis.