Measure Theory

Measure theory is a branch of mathematics that studies the concept of "measure," which can be thought of as a generalization of the intuitive idea of length, area, and volume. The foundation of measure theory is the Lebesgue measure, which is a way to assign a volume to subsets of real numbers.

1. σ-Algebra

For a given set $X$, a σ-algebra $\mathcal{F}$ on $X$ is a collection of subsets of $X$ satisfying:

• $X \in \mathcal{F}$
• If $A \in \mathcal{F}$, then its complement $A^c \in \mathcal{F}$
• If $A_1, A_2, ... \in \mathcal{F}$, then the countable union $\cup_{i=1}^{\infty}A_i \in \mathcal{F}$

2. Measure

A measure $m$ on a σ-algebra $\mathcal{F}$ over a set $X$ is a function:

$$ m: \mathcal{F} \to [0, \infty]$$

satisfying:

• $m(\emptyset) = 0$
• Countable Additivity If ${ A_i }$ is a countable collection of disjoint sets in $\mathcal{F}$, then:

$$m\left(\bigcup_{i=1}^{\infty}A_i\right) = \sum_{i=1}^{\infty} m(A_i)$$

3. Lebesgue Measure on $\mathbb{R}$

For an interval $I = [a, b]$, where $a < b$, the Lebesgue measure $m(I)$ is defined as $b - a$. The Lebesgue measure can be extended to more complicated subsets of $\mathbb{R}$ using the properties of measures.

4. Measurable Functions

A function $f: X \to \mathbb{R}$ is said to be $\mathcal{F}$-measurable if for every open set $O \subseteq \mathbb{R}$, $f^{-1}(O) \in \mathcal{F}$. In other words, the preimage of any open set in $\mathbb{R}$ should be a measurable set in $X$.

5. Integration with respect to a Measure

Given a measurable function $f$ and a measure $m$, the integral of $f$ with respect to $m$ is denoted by:

$$\int_X f dm$$

If $X = \mathbb{R}$ and $m$ is the Lebesgue measure, this is the Lebesgue integral. It's a generalization of the Riemann integral that can handle more complicated functions.

6. Convergence Theorems

These theorems connect the concepts of measure and integration:

• Monotone Convergence Theorem (MCT): If ${ f_n }$ is a sequence of non-negative measurable functions such that $f_n \uparrow f$ pointwise, then:

$$\lim_{n \to \infty} \int_X f_n dm = \int_X f dm$$

• Dominated Convergence Theorem (DCT): If ${ f_n }$ is a sequence of measurable functions that converge pointwise to $f$, and there exists an integrable function $g$ (with respect to the measure $m$) such that $|f_n| \leq g$ for all $n$, then:

$$\lim_{n \to \infty} \int_X f_n dm = \int_X f dm$$

Applications

Measure theory has applications in various areas of mathematics and its applications:

• It provides the foundation for Lebesgue integration, which extends the concept of integration to more functions than the traditional Riemann integral.
• It's central to probability theory where a probability measure is defined on a σ-algebra of events.
• It's used in functional analysis and quantum mechanics.
• Ergodic theory and dynamical systems also use ideas from measure theory.