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Basic theorems of the Lebesgue integral

The core convergence theorems: Monotone Convergence, Dominated Convergence, and Fatou's Lemma.
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The statement of the theorem

Let (Ω,F,μ)(\Omega, \mathcal{F}, \mu) be a complete measure space. Consider a sequence of measurable functions fn:Ω→[0,∞]f_n: \Omega \to [0, \infty] such that fn→ff_n \to f pointwise almost everywhere (a.e.).\n\n1. **Monotone Convergence Theorem (MCT):** If the sequence (fn)(f_n) is non-decreasing, i.e., 0≤f1≤f2≤…0 \le f_1 \le f_2 \le \dots a.e., then the limit function f=lim⁡n→∞fnf = \lim_{n\to\infty} f_n is measurable, and its integral is given by:\nlim⁡n→∞∫Ωfn dμ=∫Ωlim⁡n→∞fn dμ=∫Ωf dμ\lim_{n\to\infty} \int_{\Omega} f_n \, d\mu = \int_{\Omega} \lim_{n\to\infty} f_n \, d\mu = \int_{\Omega} f \, d\mu\n\n2. **Fatou's Lemma:** For any sequence of non-negative measurable functions (fn)(f_n), the following inequality holds:\n∫Ωlim inf⁡n→∞fn dμ≤lim inf⁡n→∞∫Ωfn dμ\int_{\Omega} \liminf_{n\to\infty} f_n \, d\mu \le \liminf_{n\to\infty} \int_{\Omega} f_n \, d\mu\n\n3. **Dominated Convergence Theorem (DCT):** If there exists a non-negative integrable function g∈L1(μ)g \in L^1(\mu) (the dominating function) such that ∣fn∣≤g|f_n| \le g a.e. for all nn, and fn→ff_n \to f pointwise a.e., then ff is integrable, and the limit and integral commute:\nlim⁡n→∞∫Ωfn dμ=∫Ωlim⁡n→∞fn dμ=∫Ωf dμ\lim_{n\to\infty} \int_{\Omega} f_n \, d\mu = \int_{\Omega} \lim_{n\to\infty} f_n \, d\mu = \int_{\Omega} f \, d\mu
Source: Wikipedia