DOMINATED CONVERGENCE THEOREM
Lebesgue's dominated convergence theorem provides sufficient conditions under which pointwise convergence of a sequence of functions implies convergence of the integrals. It's one of the reasons that makes #Lebesgue integration more powerful than #Riemann integration. The theorem an be stated as follows:

Let \((f_n)\) be a sequence of measurable functions on a measure space \((\mathcal{S},\Sigma,\mu)\). Suppose that \((f_n)\) converges pointwise to a function \(f\) and is dominated by some Lebesgue integrable function \(g\), i.e. \(|f_n(x)|\leq g(x)\ \forall n\) and \(\forall x\in\mathcal{S}\). Then, \(f\) is Lebesgue integrable, and

\[\displaystyle\lim_{n\to\infty}\int_\mathcal{S}f_n\ \mathrm{d}\mu=\int_\mathcal{S}f\ \mathrm{d}\mu\]
#ConvergenceTheorem #Convergence #DominatedConvergenceTheorem #Lebesgue #MeasurableFunction #LebesgueFunction #LebesgueIntegration #RiemannIntegration #MeasureSpace