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 $(\mathcal{S},\mathrm{\Sigma },\mu )$. Suppose that $(f_n)$ converges pointwise to a function $f$ and is dominated by some Lebesgue integrable function $g$, i.e. $|f_n(x)|\le g(x)\mathrm{\forall }n$ and $\mathrm{\forall }x\in \mathcal{S}$. Then, $f$ is Lebesgue integrable, and

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathcal{S}}{f}_n\mathrm{d}\mu ={\int }_{\mathcal{S}}f\mathrm{d}\mu }$$
#ConvergenceTheorem #Convergence #DominatedConvergenceTheorem #Lebesgue #MeasurableFunction #LebesgueFunction #LebesgueIntegration #RiemannIntegration #MeasureSpace