LebesgueIntegration

Lebesgue's approach to integration is significantly more general than the traditional Riemann approach, and it's based on the concept of measure. In the context of the real line, the Lebesgue measure $\mu$ generalizes the notion of length. The Lebesgue integral is defined for a wider class of functions than the Riemann integral and is particularly adept at handling limits and functions with discontinuities.

Key Concepts and Formulas

1. Lebesgue Measure:

   • The measure of an interval $[a, b]$ on the real line is given by $\mu([a, b]) = b - a$.

2. Simple Functions:

   • A simple function $\phi$ is defined as a finite sum:

$$\phi(x) = \sum_{i=1}^{n} a_i \chi_{A_i}(x)$$

where $a_i$ are constants, $A_i$ are measurable sets, and $\chi_{A_i}$ is the characteristic function of $A_i$.

3. Integral of Simple Functions:
   • The integral of a simple function $\phi$ over a measurable set $E$ is:

$$\int_E \phi \, d\mu = \sum_{i=1}^{n} a_i \mu(A_i \cap E)$$

4. Lebesgue Integral:
   • For a non-negative measurable function $f$, the Lebesgue integral over a set $E$ is defined as:

$$\int_E f \, d\mu = \sup \left\{ \int_E \phi \, d\mu : 0 \leq \phi \leq f, \phi \text{ is simple} \right\}$$

• For a general measurable function $f$, it is defined as:

$$\int_E f \, d\mu = \int_E f^+ \, d\mu - \int_E f^- \, d\mu$$

where $f^+$ and $f^-$ are the positive and negative parts of $f$, respectively.

5. Dominated Convergence Theorem:
   • Let ${f_n}$ be a sequence of measurable functions that converge almost everywhere to a function $f$, and let $g$ be an integrable function such that $|f_n(x)| \leq g(x)$ for all $n$ and almost every $x$. Then, $f$ is integrable and:

$$\lim_{n \to \infty} \int_E f_n \, d\mu = \int_E f \, d\mu$$

These formulas and concepts form the central concepts of Lebesgue integration, providing an elegant framework for expressing a broad array of mathematical results, especially where Riemann integration cannot be applied.