M.Sc. Mathematics – Core Theory II
Measure Theory & Lebesgue Integration
Goal of this module – Equip you with the rigorous foundations of integration on general measure spaces, and with the key analytic tools (convergence theorems, (L^p) spaces, Fubini/Tonelli) that underpin modern analysis, probability, PDEs and many other mathematical fields.
1. Simple Explanation
| Concept | What it is | Why we need it |
|---|---|---|
| Measure space ((X,\mathcal{F},\mu)) | A set (X), a σ‑algebra (\mathcal{F}) of subsets, and a measure (\mu:\mathcal{F}\to[0,\infty]) | Provides a formal way to talk about “size” of sets beyond length/area/volume |
| Measurable function | Function (f:X\to\mathbb{R}) such that ({x:f(x)>a}\in\mathcal{F}) for all (a\in\mathbb{R}) | Only measurable functions can be integrated |
| Lebesgue integral | Integral defined as the limit of integrals of simple functions approximating (f) | Extends Riemann integral, works for a far larger class of functions, and behaves well under limits |
| (L^p) spaces | Sets of measurable functions with finite (p)-norm (|f|_p=(\int | f |
| Convergence theorems | Monotone Convergence Theorem (MCT), Dominated Convergence Theorem (DCT), Fatou’s Lemma | Give conditions under which we can interchange limits and integrals – essential for analysis |
2. Key Points
2.1 Fundamental Definitions
| Symbol | Definition | Notes |
|---|---|---|
| ((X,\mathcal{F},\mu)) | Measure space | (\mu(\emptyset)=0), countable additivity |
| (\mathcal{F}) | σ‑algebra | Closed under complements and countable unions |
| Simple function | (s(x)=\sum_{k=1}^n a_k \mathbf{1}_{A_k}(x)) with (A_k\in\mathcal{F}) | Base of Lebesgue integration |
| (\int_X s,d\mu = \sum_{k=1}^n a_k \mu(A_k)) | Integral of a simple function | |
| Measurable function | (f:X\to\mathbb{R}) with ({x: f(x)>a}\in\mathcal{F}) for all (a) | Equivalent to preimage of Borel sets being measurable |
| Lebesgue integral of (f) | (\int_X f,d\mu = \sup{\int_X s,d\mu: 0\le s\le f,\ s\ \text{simple}}) | Requires (f\ge0). For arbitrary (f), split into positive/negative parts. |
| (L^p(X,\mu)) | ({f\text{ measurable}: \int | f |
2.2 Core Theorems
| Theorem | Statement | Consequences |
|---|---|---|
| Monotone Convergence Theorem (MCT) | If (0\le f_n\uparrow f) pointwise, then (\int f_n d\mu \uparrow \int f d\mu). | Allows limit inside integral for increasing sequences. |
| Fatou’s Lemma | (\int \liminf f_n \le \liminf \int f_n). | Provides lower bound; useful for proving MCT. |
| Dominated Convergence Theorem (DCT) | If (f_n\to f) a.e. and ( | f_n |
| Tonelli’s Theorem | For non‑negative measurable (f) on product space, (\int_{X\times Y} f d(\mu\times\nu)=\int_X(\int_Y f(x,y)d\nu(y))d\mu(x)). | Allows iterated integrals for non‑negative functions. |
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