📏Geometric Measure Theory Unit 1 – Introduction to Measure Theory

Key Concepts and Definitions

• Measure theory extends the notion of length, area, and volume to more abstract spaces
• A measure is a function that assigns a non-negative real number or $+\infty$ to subsets of a set
• Measures must satisfy countable additivity: the measure of a countable union of disjoint sets is the sum of their individual measures
• Measurable sets are the domain of a measure and form a $\sigma$-algebra
• Measurable functions are functions between measurable spaces that preserve measurability
• Integration with respect to a measure generalizes Riemann integration
  • Lebesgue integration is a key example that integrates measurable functions with respect to Lebesgue measure
• Convergence theorems (Monotone Convergence Theorem, Dominated Convergence Theorem) provide conditions for interchanging limits and integrals
• Product measures (Fubini's Theorem) allow for integration over product spaces

Measure Spaces and σ-Algebras

• A measurable space is a pair $(X, \mathcal{A})$ where $X$ is a set and $\mathcal{A}$ is a $\sigma$-algebra on $X$
• A $\sigma$-algebra $\mathcal{A}$ on a set $X$ is a collection of subsets of $X$ that satisfies:
  • $X \in \mathcal{A}$
  • If $A \in \mathcal{A}$, then $A^c \in \mathcal{A}$ (closure under complements)
  • If $A_n \in \mathcal{A}$ for $n \in \mathbb{N}$, then $\bigcup_{n=1}^{\infty} A_n \in \mathcal{A}$ (closure under countable unions)
• The elements of a $\sigma$-algebra are called measurable sets
• The Borel $\sigma$-algebra on a topological space $X$ is the smallest $\sigma$-algebra containing all open sets
• A measure space is a triple $(X, \mathcal{A}, \mu)$ where $(X, \mathcal{A})$ is a measurable space and $\mu$ is a measure on $\mathcal{A}$
• A probability space is a measure space $(X, \mathcal{A}, P)$ where $P(X) = 1$

Lebesgue Measure on R^n

• Lebesgue measure $\lambda$ is a complete measure on $\mathbb{R}^n$ that extends the notion of length, area, and volume
• For intervals $I = [a_1, b_1] \times \cdots \times [a_n, b_n]$ in $\mathbb{R}^n$, the Lebesgue measure is defined as $\lambda(I) = \prod_{i=1}^n (b_i - a_i)$
• Lebesgue measure is translation invariant: $\lambda(A + x) = \lambda(A)$ for all measurable sets $A$ and $x \in \mathbb{R}^n$
• Lebesgue measure is countably additive: for a countable collection of disjoint measurable sets $\{A_i\}$, $\lambda(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} \lambda(A_i)$
• Lebesgue measure is complete: if $A$ is measurable and $\lambda(A) = 0$, then any subset of $A$ is also measurable
• Lebesgue measure is the unique complete translation invariant measure on $\mathbb{R}^n$ that normalizes the unit cube to have measure 1

Measurable Functions and Integration

• A function $f: X \to Y$ between measurable spaces $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ is measurable if $f^{-1}(B) \in \mathcal{A}$ for all $B \in \mathcal{B}$
• Simple functions are measurable functions that take on finitely many values
  • Any measurable function can be approximated by a sequence of simple functions
• The Lebesgue integral of a non-negative measurable function $f$ on a measure space $(X, \mathcal{A}, \mu)$ is defined as $\int_X f \, d\mu = \sup \left\{ \int_X s \, d\mu : 0 \leq s \leq f, s \text{ simple} \right\}$
• For a general measurable function $f$, the Lebesgue integral is defined as $\int_X f \, d\mu = \int_X f^+ \, d\mu - \int_X f^- \, d\mu$, provided at least one of the integrals on the right is finite
• Lebesgue integration extends Riemann integration and has better properties (integrable functions form a complete normed vector space)

Convergence Theorems

• The Monotone Convergence Theorem states that if $\{f_n\}$ is a sequence of non-negative measurable functions on a measure space $(X, \mathcal{A}, \mu)$ with $f_n \leq f_{n+1}$ for all $n$ and $f = \lim_{n \to \infty} f_n$, then $\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu$
• The Dominated Convergence Theorem states that if $\{f_n\}$ is a sequence of measurable functions on a measure space $(X, \mathcal{A}, \mu)$ with $f_n \to f$ pointwise and there exists an integrable function $g$ such that $|f_n| \leq g$ for all $n$, then $\lim_{n \to \infty} \int_X f_n \, d\mu = \int_X f \, d\mu$
• Fatou's Lemma states that if $\{f_n\}$ is a sequence of non-negative measurable functions on a measure space $(X, \mathcal{A}, \mu)$, then $\int_X \liminf_{n \to \infty} f_n \, d\mu \leq \liminf_{n \to \infty} \int_X f_n \, d\mu$
• These theorems provide conditions under which limits and integrals can be interchanged, which is crucial for many applications

Product Measures and Fubini's Theorem

• Given two measure spaces $(X, \mathcal{A}, \mu)$ and $(Y, \mathcal{B}, \nu)$, the product measure $\mu \times \nu$ on the product space $X \times Y$ is defined by $(\mu \times \nu)(A \times B) = \mu(A) \nu(B)$ for measurable rectangles $A \times B$
• Fubini's Theorem states that for a measurable function $f$ on the product space $X \times Y$, $\int_{X \times Y} f(x, y) \, d(\mu \times \nu) = \int_X \left( \int_Y f(x, y) \, d\nu(y) \right) \, d\mu(x) = \int_Y \left( \int_X f(x, y) \, d\mu(x) \right) \, d\nu(y)$
  • This allows for the computation of integrals over product spaces by iterating integrals
• Tonelli's Theorem is a version of Fubini's Theorem for non-negative measurable functions that does not require integrability
• These theorems are fundamental for multivariable integration and have applications in probability theory and other areas

Applications in Geometric Analysis

• Measure theory provides a rigorous foundation for the study of volume, surface area, and other geometric quantities
• The Hausdorff measure generalizes Lebesgue measure and allows for the measurement of lower-dimensional sets (fractals, manifolds)
• The co-area formula relates the integral of a function over a set to the integral of its restriction to level sets, weighted by the Hausdorff measure of the level sets
  • This is a key tool in geometric measure theory and has applications in the study of minimal surfaces and isoperimetric problems
• The Brunn-Minkowski inequality relates the measures of sets and their Minkowski sums, providing a fundamental link between measure theory and convex geometry
• Measure-theoretic techniques are used to study the regularity of minimal surfaces and to prove existence results for variational problems in geometry

Common Pitfalls and Tips

• Remember that not all subsets of a measure space are measurable (Vitali set)
• Be careful when applying convergence theorems; check that the hypotheses are satisfied
• Measurability of functions is not always intuitive; continuous functions are measurable, but not all measurable functions are continuous
• When working with product measures, be sure to use the correct product $\sigma$-algebra
• In applications, pay attention to the choice of measure and whether it is appropriate for the problem at hand
• Many important results in measure theory and integration have subtle hypotheses; make sure to understand and verify these conditions when using the theorems
• Measure theory can be abstract and challenging at first; working through concrete examples and counterexamples can help build intuition