Measure Theory

Area

Measure theory attempts to extend the concepts of length, area, and volume to arbitrary subsets in ℝ, ℝ2, and ℝ3. In general it attempts to assign an n-volume to arbitrary subsets of ℝn.

Consider area in ℝ2. The unit square is assigned an area of 1. The area of other figures is defined by how many unit squares they contain.

A rectangle with sides of length 2 and 3 can be chopped into 6 unit squares, so we get an area of 6 for the rectangle. When reasoning thus we are using finite additivity: i.e. when the sets Ej are disjoint:

(1)
\begin{align} \mu\big( \bigcup_1^n E_j\big) = \sum_1^n \mu(E_j) \end{align}

We are also assuming the area of the unit square doesn't change when we move it. That is, area is invariant under translation. We also take area to be invariant under rotation and reflection.

If a rectangle has sides of length 1/2 and 1/3, we can show that has area 1/6 by chopping up the unit square into six parts.

area.png

If a rectangle has sides of length 5/3 and 3/2, we can show that it has area 15/6 by chopping it into 15 rectangles with sides of length 1/2 and 1/3.

Proceeding in this fashion, we can convince ourselves that the area of a rectangle with sides of length a and b is ab, at least when a and b are rational.

parallelogram.png

If a parallelgram has opposite sides with length b and the distance between those two sides is h, then the area is bh. We show this by chopping off a triangle from one side to "square off" one end, and translate the triangle to the other end to square it off as well, converting the parallelogram to a rectangle with equal area.

triangle.png

We can construct a parallelogram with twice the area of any triangle by duplicating the triangle and rotating the copy. From this we show that a triangle has area bh/2, where b is the length of one side and h is the distance from that side to the vertex not on the side.

If we know the coordinates of the vertices of a triangle, then b is easy to determine using the distance formula for two points.

h is the distance of the other vertex (x0, y0) to the line containing the opposite side. If the other two vertices are (x1, y1) and (x2, y2), the distance is:

(2)
\begin{align} \frac{\big|x_0(y_2 - y_1) - y_0(x_2 - x_1) + x_2 y_1 - y_2 x_1\big|}{\sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2}} \end{align}

Knowing the the area of a triangle, we can in principle determine the area of any region bound by a polygon with rational coordinates using polygon triangulation.

Archimedes determined the area of a circle using an infinite sequence of inscribed polygons. The technique requires a stronger condition than finite additivity, namely countable additivity. Countable additivity also allows us to extend our formulas for rectangles and triangles to figures with irrational side lengths.

The discovery of the Fundamental Theorem of Calculus in the 17th century made it possible to find the area of figures bounded by curves for which we can find the antiderivative. And even if we can't find a closed form form for the antiderivative, we can integrate a Taylor series representation of the curve directly.

The proof that Isaac Barrow gave for the FTC was geometric in nature. As far as I know, the first analytic proof depended on Riemann sums and is outlined below.

A partition P of [a, b] is a finite set of points x0, x1, … xn where

(3)
\begin{align} a = x_0 \leq x_1 \leq \cdots \leq x_{n-1} \leq x_n = b \end{align}

We define Δxi = xi - xi-1 and we also make these definitions:

(4)
\begin{align} &M_i = \sup f(x) \;\;\;\; (x_{i-1} \leq x \leq x_i) \\ &m_i = \inf f(x) \;\;\;\; (x_{i-1} \leq x \leq x_i) \\ &U(P, f) = \sum_{i=1}^n M_i \Delta x_i \\ &L(P, f) = \sum_{i=1}^n m_i \Delta x_i \\ \end{align}

The upper and lower Riemann integrals are:

(5)
\begin{align} &\overline{\int}_a^b f \; dx = \inf U(P, f) \\ &\underline{\int}_a^b f \; dx = \sup L(P, f) \end{align}

If the upper and lower integrals are the same, then f is Riemann integrable on [a, b].

The FTC states that if f is Riemann integrable on [a, b] and if there is a differentiable function F on [a, b] such that F' = f, then

(6)
\begin{align} \int_a^b f(x) \; dx = F(b) - F(a) \end{align}

The proof fixes ε > 0 and chooses a partition P of [a, b] such that U(P, f) - L(P, f) < ε. It uses the MVT to find ti
in [xi-1, x,,i] such that

(7)
\begin{align} F(x_i) - F(x_{i-1}) = f(t_i) \Delta x_i \end{align}

from which it follows that

(8)
\begin{align} \bigg|F(b) - F(a) - \int_a^b f(x) \; dx \bigg| < \epsilon \end{align}

Students of analysis often wonder why, after learning the Riemann integral, they must next turn their attention to the Lebesgue integral, which is conceptually more difficult. The Lebesgue integral allows us to the answer the question of which functions are Riemann integrable: those which are continuous almost everywhere. The Lebesgue integral also allows us to prove results such as the Monotone Convergence Theorem and the Dominated Convergence theorem which allow us to switch the order of integration and limits in an expression.

Definitions

  • algebra
  • σ-algebra
  • generated σ-algebra
  • Borel σ-algebra
  • Borel set
  • Gδ set
  • Fσ set
  • product σ-algebra
  • elementary family
  • countable additivity property
  • measure
  • finite measure
  • σ-finite measure
  • semifinite measure
  • null set
  • true almost everywhere
  • complete measure
  • outer measure
  • μ*-measurable
  • premeasure
  • Lebesgue-Stieltjes measure
  • Lebesgue measure
  • measurable function
  • characteristic function
  • simple function
  • standard representation of a simple function
  • integral of a simple function
  • integral of a nonnegative function
  • positive and negative parts of a function
  • integral of a real valued function
  • integral of a complex valued function
  • Lebesgue integral
  • Riemann integral
  • uniform convergence
  • pointwise convergence
  • a.e. convergence
  • Cauchy sequence in measure
  • convergence in measure
  • measurable rectangle
  • product measure
  • x-section
  • y-section
  • monotone class
  • generated monotone class
  • signed measure
  • positive measure
  • extended μ-integrable function
  • positive set
  • negative set
  • null set (of a signed measure)
  • Hahn decomposition
  • mutually singular
  • Jordan decomposition
  • positive variation
  • negative variation
  • total variation
  • finite signed measure
  • σ-finite signed measure

An algebra is a set 𝒜 of subsets of X that is closed under finite unions and complements.

An σ-algebra is an algebra that is closed under countable unions.

For any set ℰ ⊆ P(X), there is a unique smallest σ-algebra containing ℰ which is called the σ-algebra generated by ℰ.

The σ-algebra generated by the open sets in a metric space or a topological space X is called the Borel σ-algebra of X and is denoted ℬX.

An element of a Borel σ-algebra is called a Borel set.

A countable intersection of open sets is called a Gδ set.

A countable union of closed sets is called an Fσ set.

Let {Xα}αA be an indexed collection of nonempty sets, X = ∏αAXα, and πα: XXα the coordinate maps. The product σ-algebra is the σ-algebra generated by

(9)
\begin{align} \{\pi_\alpha^{-1} (E_\alpha): E_\alpha \in \mathcal{M}_\alpha, \; \alpha \in A\} \end{align}

An elementary family is a collection ℰ of subsets of of X such that (1) ∅ ∈ ℰ, (2) if E, F ∈ ℰ then EF ∈ ℰ, and (3) if E ∈ ℰ then EC is a finite disjoint union of members of ℰ.

Let X be a set with σ-algebra ℳ. A measure on (X, ℳ) is a function μ:ℳ → [0, ∞] such that (i) μ(∅) = 0 and (ii) μ has the countable additivity property, which states that if

(10)
\begin{align} \{E_j\}_1^\infty \end{align}

is a sequence of disjoint sets in ℳ, then

(11)
\begin{align} \mu\big(\bigcup_1^\infty E_j\big) = \sum_1^\infty \mu(E_j) \end{align}

(X, ℳ, μ) is called a measurable space, and ℳ contains the measurable sets.

μ is finite if for all X ∈ ℳ, μ(X) < ∞.

μ is σ-finite if for all X ∈ ℳ there exists a countable set of Ei in ℳ such that

(12)
\begin{align} X = \bigcup_1^\infty E_i \end{align}

and μ(Ei) < ∞ for all Ei.

μ is semifinite if for each E ∈ ℳ such that μ(E) = ∞ there exists FE such that 0 < μ(F) < ∞.

A set E such that μ(E) = 0 is called a null set.

A statement which is true for all points except those in a null set is said to be true almost everywhere.

A measure is complete if the σ-algebra contains all subsets of null sets.

An outer measure for a nonempty set X is a function μ*: 𝒫(X) → [0, ∞] that satisfies

(13)
\begin{align} \mu^*(\emptyset) = 0 \end{align}
(14)
\begin{align} \mu^*(A) \leq \mu^*(B) \;\;\mathrm{if}\;\; A \subseteq B \end{align}

Let μ* be an outer measure. A set A is μ*-measurable if for all EX

(15)
\begin{align} \mu^*(E) = \mu^*(E \cap A) + \mu^*(E \cap A^C) \end{align}

If 𝒜 ⊆ 𝒫(X) is an algebra, then μ0: 𝒜 → [0, ∞] is a premeasure if

(16)
\begin{align} \mu_0(\emptyset) = 0 \; \text{and if} \; \{A_j\}_1^\infty \; \text{is a sequence of disjoint sets in} \; \mathcal{A} \; \text{such that} \; \bigcup_1^\infty A_j \in \mathcal{A} \; \text{then} \; \mu_0\big(\bigcup_1^\infty A_j \big) = \sum_1^\infty \mu_0(A_j) \end{align}

Let F: ℝ → ℝ be an increasing, right-continuous function. The Lebesgue-Stieltjes measure is the unique measure μF such that μF((a, b]) = F(b) - F(a).

The Lebesque measure is the Lebesgue-Stieltjes measure for the function F(x) = x.

If (X, ℳ) and (Y, 𝒩) are measurable spaces, then f: XY is (ℳ, 𝒩)-measurable if f-1(E) ∈ ℳ for all E ∈ 𝒩.

For any EX, the characteristic function χE of E is defined as

(17)
\begin{align} \chi_E(x) = \begin{cases} 1 & x \in E \\ 0 & x \notin E \end{cases} \end{align}

A simple function on X is a finite linear combination, with complex coefficients, of characteristic functions of sets in ℳ.

The standard representation of a simple function is the form

(18)
\begin{align} f = \sum_1^n z_j \,\chi_{E_j} \end{align}

where Ej = f-1({z_j}) and range(f) = {z1, …, zn}.

L+ is the space of measurable functions from X to [0, ∞].

Let φ be a simple function in the space of measurable functions from X to [0, ∞] with standard representation

(19)
\begin{align} \phi = \sum_1^n a_j \chi_{E_j} \end{align}

The integral of φ is:

(20)
\begin{align} \int \phi \,d \mu = \sum_1^n a_j \, \mu (E_j) \end{align}

In the above definition, the convention that 0 ⋅ ∞ = 0 is observed.

If f is any measurable function from X to [0, ∞], the integral is:

(21)
\begin{align} \int f \, d\mu = \sup \bigg\{ \int \phi \, d\mu : 0 \leq \phi \leq f, \;\; \phi \;\; \mathrm{simple} \bigg\} \end{align}

The extended real number system, [-∞, ∞], is a topological space consisting of the real numbers plus points for infinity and negative infinity.

Let f: X → [-∞, ∞]. The positive part of f is

(22)
\begin{align} f^+(x) = \max (f(x), \, 0) \end{align}

The negative part of f is

(23)
\begin{align} f^-(x) = \max (-f(x), \, 0) \end{align}

The integral of a real valued function f is defined as

(24)
\begin{align} \int f = \int f^+ - \int f^- \end{align}

However, the integral is undefined when both ∫f+ and ∫f- are infinite.

If f is complex valued and if ∫E|f| < ∞, then the integral of f is

(25)
\begin{align} \int f = \int \mathrm{Re}\;f + i \int \mathrm{Im}\;f \end{align}

The space of complex valued functions f for which ∫E|f| < ∞ is denoted L1.

The Lebesgue integral is the integral as defined above when the measure is the Lebesgue measure on ℝ.

The Riemann integral

A sequence of functions {fn} defined on X converges uniformly to f if for every ε > 0 there exists N such that when n > N, |fn(x) - f(x)| < ε for all xX.

A sequence of functions {fn} defined on X converges pointwise to f if for every xX and every ε > 0 there exists N which may depend on x such that when n > N, |fn(x) - f(x)| < ε.

A sequence of functions {fn} defined on X converges a.e. to f if it converges pointwise to f for all points of X except for points in a set of measure zero.

Cauchy sequence in measure

convergence in measure

If (X, ℳ, μ) and (Y, 𝒩, ν) are measurable spaces, then a measurable rectangle is set in the product σ-algebra ℳ ⊗ 𝒩 of the form A × B, where A ∈ ℳ and B ∈ 𝒩.

If (X, ℳ, μ) and (Y, 𝒩, ν) are measurable spaces, then the product measure μ × ν is the restriction to ℳ ⊗ 𝒩 of the outer measure generated by the measurable rectangles of X × Y that extends the premeasure

(26)
\begin{align} \pi(E) = \sum_1^n \mu(A_j) \nu(B_j) \end{align}

If (X, ℳ) and (Y, 𝒩) are measurable spaces and EX × Y, the x-section Ex is the set {yY: (x, y) ∈ E}.

Similarly if (X, ℳ) and (Y, 𝒩) are measurable spaces and EX × Y, the y-section Ey is the set {xX: (x, y) ∈ E}.

A monotone class on a space X is a subset 𝒞 of 𝒫(X) that is closed under countable increasing unions and countable decreasing intersections.

For any ℰ ⊆ 𝒫(X) there is a unique smallest monotone class containing ℰ, which we call the monotone class generated by ℰ.

If (X, ℳ) is a measurable space, a signed measure is a function ν: ℳ → [-∞, ∞] such that (1) ν(∅) = 0, (2) ν assumes at most of the values ±∞, and (3) if {Ej} is a sequence of disjoint sets in ℳ, then

(27)
\begin{align} \nu\bigg(\bigcup_1^\infty E_j\bigg) = \sum_1^\infty \nu(E_j) \end{align}

where the latter sum converges absolutely if

(28)
\begin{align} \nu\bigg(\bigcup_1^\infty E_j\bigg) \end{align}

is finite.

A positive measure is a signed measure with range [0, ∞]. This is a synonym for a measure.

f is an extended μ-integrable function if at least one of ∫f + dμ and ∫f - dμ is finite.

If ν is a signed measure on (X, ℳ), then E ∈ ℳ is a positive set if ν(F) ≥ 0 for all FE. If ν(F) ≤ 0 or ν(F) = 0 for all FE then E is a negative set or null set, respectively. Compare with null sets for positive measures defined previously.

The Hahn decomposition of a signed measure is the disjoint pair of sets P and N such that PN = X, P is positive, and N is negative. The Hahn Decomposition Theorem guarantees that a Hahn decomposition exists.

Two measures μ and ν on a meaurable space (X, ℳ, μ) are mutually singular if there exist disjoint E and F such that EF = X, E is null for μ, and F is null for ν.

The Jordan decomposition of a signed measure ν is a pair of positive measures ν+ and ν- such that ν = ν+ - ν- and ν+ν-. The Jordan Decomposition Theorem guarantees that a Jordan decomposition exists.

The ν+ and ν- in the Jordan decomposition of a signed measure ν are called the positive variation and negative variation of ν.

The total variation |ν| of a signed measure ν is ν+ + ν- where ν+ and ν- are from the Jordan decomposition. A signed measure is finite if its total variation is finite. Similarly a signed measure σ-finite if its total variation is σ-finite.

If ν is a signed measure and μ a positive measure on (X, ℳ), then ν is absolutely continuous with respect to μ, written ν « μ, if ν(E) = 0 for every E ∈ ℳ for which μ(E) = 0.

Lebesgue decomposition

Radon-Nikodym derivative

TODO

  • Lebesgue measure
  • Lebesgue integral
  • Lebesgue-Stieltjes measure
  • Lebesgue-Stieltjes integral
  • Riemann Integral
  • types of convergence
  • product  σ-algebras
  • product measures

Theorems

Monotonicity

If (X, ℳ, μ) is a measure space, E, F ∈ ℳ, and EF, then μ(E) ≤ μ(F).

Subadditivity

If (X, ℳ, μ) is a measure space and

(29)
\begin{align} \{E_j\}_1^\infty \subseteq \mathcal{M} \end{align}

then

(30)
\begin{align} \mu\big(\bigcup_1^\infty E_j\big) \leq \sum_1^\infty \mu(E_j) \end{align}

Continuity from below

If (X, ℳ, μ) is a measure space,

(31)
\begin{align} \{E_j\}_1^\infty \subseteq \mathcal{M} \end{align}

and E1 ⊆ E2 ⊆ ⋯, then

(32)
\begin{align} \mu\big(\bigcup_1^\infty E_j\big) = \lim_{j \rightarrow \infty} \mu(E_j) \end{align}

Continuity from above

If (X, ℳ, μ) is a measure space,

(33)
\begin{align} \{E_j\}_1^\infty \subseteq \mathcal{M} \end{align}

and E1 ⊇ E2 ⊇ ⋯, then

(34)
\begin{align} \mu\big(\bigcap_1^\infty E_j\big) = \lim_{j \rightarrow \infty} \mu(E_j) \end{align}

Carathéodory's Theorem

If μ* is an outer measure on X, the collection ℳ of μ*-measurable sets is a σ-algebra, and the restriction of μ* to ℳ is a complete measure.

Monotone Convergence

If {fn} is a sequence in L+ such that fjfj+1 for all j, and

(35)
\begin{align} f = \lim_{n \rightarrow \infty} f_n \;\; (= \sup_n f_n) \end{align}

then

(36)
\begin{align} \int f = \lim_{n \rightarrow \infty} \int f_n \end{align}

Fatou's Lemma

If {fn} is any sequence in L+, then

(37)
\begin{align} \int( \lim \inf f_n) \leq \lim \inf \int f_n \end{align}

Dominated Convergence Theorem

Let {fn} be a sequence in L1 such that (a) fnf a.e., and (b) there exists a nonnegative gL1 such that |fn| ≤ g a.e. for all n. Then fL1 and ∫ f = limn → ∞fn.

Egoroff's Theorem

Suppose that μ(X) < ∞ and f1, f2, … and f are measurable complex-valued functions on X and fnf a.e. Then for every ε > 0 there exists EX such that μ(E) < ε and fnf uniformly on EC.

Monotone Class Lemma

If 𝒜 is an algebra of subsets of X, then the monotone class 𝒞 generated by 𝒜 coincides with the σ-algebra ℳ generated by 𝒜.

Fubini-Tonelli

Suppose that (X, ℳ, μ) and (Y, 𝒩, ν) are σ-finite measure spaces.

a. (Tonelli) If fL+(X × Y), then the functions g(x) = ∫ fx dν and h(y) = ∫ fy dμ are in L+(X) and L+(Y), respectively, and

(38)
\begin{align} \int f \; d(\mu \times \nu) &= \int \bigg[ \int f(x, y) \; d\nu(y) \bigg] d\mu(x) \\ &= \int \bigg[\int f(x, y) \; d\mu(x) \bigg] d\nu(y) \end{align}

b. (Fubini) If fL1(μ × ν), then fxL1(ν) for a.e. xX, fyL1 for a.e. yY, the a.e.-defined functions g(x) = ∫ fx dν and h(y) = ∫ fydμ are in L1(μ) and L1(ν), respectively, and the equations in (a) hold.

Hahn Decomposition

If ν is a signed measure on (X, ℳ), there exists a positive set P and a negative set N for ν such that PN = X and PN = ∅. If P', N' is another such pair, then P Δ P' (= N Δ N') is null for ν.

Jordan Decomposition

If ν is a signed measure, there exist unique positive measures ν+ and ν- such that ν = ν+ - ν- and ν+ν-.

Lebesgue-Radon-Nikodym Theorem

Let ν be a σ-finite signed measure and μ a σ-finite positive measure on (X, ℳ). There exist unique σ-finite signed measures λ, ρ on (X, ℳ) such that

(39)
\begin{align} \lambda \perp \mu, \;\;\; \rho ≪ \mu, \;\;\; \nu = \lambda + \rho \end{align}

Moreover, there is an extended μ-integrable function f: X → ℝ such that dρ = f dμ, and any two such functions are equal μ-a.e.

Maximal

Lebesgue Differentiation

Fundamental Theorem of Calculus

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