Vitali Convergence Theorem

The Vitali Convergence Theorem provides conditions under which a sequence of functions converges properly when integrated. Specifically, if a sequence of functions converges everywhere to a limit function and is uniformly controlled in a way that prevents large fluctuations (called uniform integrability), then the integral (area under the curve) of the functions approaches the integral of the limit function. This theorem is essential in analysis, ensuring that pointwise convergence combined with controlled behavior allows us to exchange limits and integrals reliably, which is vital for advanced calculus and probability theory applications.