Pustam | पुस्तम | পুস্তম🇳🇵<p>DOMINATED CONVERGENCE THEOREM<br>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 <a href="https://mathstodon.xyz/tags/Lebesgue" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Lebesgue</span></a> integration more powerful than <a href="https://mathstodon.xyz/tags/Riemann" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Riemann</span></a> integration. The theorem an be stated as follows:</p><p>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</p><p>\[\displaystyle\lim_{n\to\infty}\int_\mathcal{S}f_n\ \mathrm{d}\mu=\int_\mathcal{S}f\ \mathrm{d}\mu\]<br><a href="https://mathstodon.xyz/tags/ConvergenceTheorem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>ConvergenceTheorem</span></a> <a href="https://mathstodon.xyz/tags/Convergence" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Convergence</span></a> <a href="https://mathstodon.xyz/tags/DominatedConvergenceTheorem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>DominatedConvergenceTheorem</span></a> <a href="https://mathstodon.xyz/tags/Lebesgue" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Lebesgue</span></a> <a href="https://mathstodon.xyz/tags/MeasurableFunction" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>MeasurableFunction</span></a> <a href="https://mathstodon.xyz/tags/LebesgueFunction" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>LebesgueFunction</span></a> <a href="https://mathstodon.xyz/tags/LebesgueIntegration" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>LebesgueIntegration</span></a> <a href="https://mathstodon.xyz/tags/RiemannIntegration" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>RiemannIntegration</span></a> <a href="https://mathstodon.xyz/tags/MeasureSpace" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>MeasureSpace</span></a></p>