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Integration Theory (Lebesgue) (Definition)

The theory of integrals, dealing with definitions, properties, and convergence theorems (e.g., Lebesgue integration).
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The statement of the theorem

Let (X,M,μ)(X, \mathcal{M}, \mu) be a complete measure space. For a measurable function f:X→Rf: X \to \mathbb{R}, we define the positive and negative parts of ff as f+=max⁡(f,0)f^+ = \max(f, 0) and f−=max⁡(−f,0)f^- = \max(-f, 0). The integral of ff with respect to the measure μ\mu, denoted ∫fdμ\int f d\mu, is defined by the following steps:\n\n1. **Simple Functions:** For a non-negative simple function ϕ=∑i=1naiχAi\phi = \sum_{i=1}^{n} a_i \chi_{A_i}, where ai≥0a_i \ge 0 are constants and Ai∈MA_i \in \mathcal{M}, the integral is defined as: \int \phi d\mu = \sum_{i=1}^{n} a_i \mu(A_i).\n2. **Non-negative Measurable Functions:** For any non-negative measurable function f: X \to [0, \infty],theLebesgueintegralisdefinedasthesupremumoverallsimplefunctions, the Lebesgue integral is defined as the supremum over all simple functions \phisuchthat such that 0 \le \phi \le f: \int f d\mu = \sup \left\{ \int \phi d\mu \mid \phi \text{ is simple and } 0 \le \phi \le f \right\}.\n3. **General Measurable Functions:** For a general measurable function f,theintegralisdefinedas:∫fdμ=∫f+dμ−∫f−dμ, the integral is defined as: \int f d\mu = \int f^+ d\mu - \int f^- d\mu. \n\nThis definition ensures that the integral is countably additive and satisfies the Monotone Convergence Theorem (MCT) and the Dominated Convergence Theorem (DCT).