----- > [!proposition] Corollary. ([[limits and integrals commute given uniform convergence for finite measure spaces]]) > > Suppose $(X, \Sigma, \mu)$ is a [[finite measure|finite]] [[measure|measure space]] (i.e. $\mu(X)<\infty$). > Suppose $(f_{n})$ is a [[sequence]] of functions [[Lp-norm|in]] $\mathcal{L}^{1}(\mu)$ [[uniform convergence|converging uniformly]] to $f:X \to \mathbb{R}$. > > Then $\lim_{n \to \infty}\int f_{n}\, d\mu =\int f \, d\mu .$ > > > [!note] Note. > Even though it is phrased here measure-theoretically, this result does not really make use of the measure-theoretic formalism's power in the same way [[monotone convergence theorem for nonnegative measurable functions|MCT]], [[Dominated Convergence Theorem|DCT]], or [[Fatou's Lemma]] do. Indeed, it is taught and heavily used in the context of [[Riemann integral|Riemann integration]] on $\mathbb{R}^{n}$. ^note > [!proposition] Proposition. (Series swap integrals given uniform convergence on finite measure spaces) > If $(g_{n}(z))$ is a [[series]] of [[continuous]] functions on $[a,b]$ whose [[sequence]] of [[partial sum]]s [[uniform convergence|converges uniformly]] in $x \in [a,b]$, then $\int _{a}^b \sum_{n=1}^{\infty} g_{n}(x) \ dx = \sum_{n=1}^{\infty} \int _{a}^b g_{n}(x) \ dx.$ > [!proof]+ Proof. ([[limits and integrals commute given uniform convergence for finite measure spaces]]) > > Using the basic [[integral]] property $|\int _{X} f \, d\mu| \leq \mu(X) \sup_{X}(f)$, we have $\begin{align} \int f_{n} \, d\mu - \int f \, d\mu \leq > \left|\int f_{n} \, d\mu - \int f \, d\mu \right| > & \leq \mu(X) \sup_{x \in X} |f_{n}(x)-f(x)| \\ > &= \mu(X) \|f_{n}-f\|_{\infty}. > \end{align}$ > Recalling that [[uniform convergence]] is equivalent to [[converge|convergence]] in the [[uniform topology|uniform (sup) topology]] on $\mathbb{R}^{X}$, taking $n \to \infty$ sends $\|f_{n}-f\|_{\infty} \to 0$, and hence $\lim_{n \to \infty} ( \int f_{n} \, d\mu - \int f \, d\mu )=0,$ > from which the result follows. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```