Table of Contents

Area
Measure theory attempts to extend the concepts of length, area, and volume to arbitrary subsets in ℝ, ℝ^{2}, and ℝ^{3}. In general it attempts to assign an nvolume to arbitrary subsets of ℝ^{n}.
Consider area in ℝ^{2}. 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 E_{j} are disjoint:
(1)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 of equal size.
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.
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.
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 (x_{0}, y_{0}) to the line containing the opposite side. If the other two vertices are (x_{1}, y_{1}) and (x_{2}, y_{2}), the distance is:
(2)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 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 x_{0}, x_{1}, … x_{n} where
(3)We define Δx_{i} = x_{i}  x_{i1} and we also make these definitions:
(4)The upper and lower Riemann integrals are:
(5)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)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 t_{i}
in [x_{i1}, x,,i] such that
from which it follows that
(8)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
 σalgebra
 generated σalgebra
 Borel σalgebra
 Borel set
 G_{δ} set
 F_{σ} set
 product σalgebra
 elementary family
 countable additivity property
 measure
 finite measure
 σfinite measure
 semifinite measure
 null set
 true almost everywhere
 complete measure
 outer measure
 μ^{*}measurable
 premeasure
 LebesgueStieltjes 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
 xsection
 ysection
 monotone class
 generated monotone class
 signed measure
 positive measure
 extended μintegrable 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
 σfinite signed measure
An algebra is a set 𝒜 of subsets of X that is closed under finite unions and complements.
An σalgebra is an algebra that is closed under countable unions.
For any set ℰ ⊆ P(X), there is a unique smallest σalgebra containing ℰ which is called the σalgebra generated by ℰ.
The σalgebra generated by the open sets in a metric space or a topological space X is called the Borel σalgebra of X and is denoted ℬ_{X}.
An element of a Borel σalgebra is called a Borel set.
A countable intersection of open sets is called a G_{δ} set.
A countable union of closed sets is called an F_{σ} set.
Let {X_{α}}_{α∈A} be an indexed collection of nonempty sets, X = ∏_{α∈A}X_{α}, and π_{α}: X → X_{α} the coordinate maps. The product σalgebra is the σalgebra generated by
(9)An elementary family is a collection ℰ of subsets of of X such that (1) ∅ ∈ ℰ, (2) if E, F ∈ ℰ then E ∩ F ∈ ℰ, and (3) if E ∈ ℰ then E^{C} is a finite disjoint union of members of ℰ.
Let X be a set with σalgebra 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)is a sequence of disjoint sets in ℳ, then
(11)(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 σfinite if for all X ∈ ℳ there exists a countable set of E_{i} in ℳ such that
(12)and μ(E_{i}) < ∞ for all E_{i}.
μ is semifinite if for each E ∈ ℳ such that μ(E) = ∞ there exists F ⊆ E 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 σalgebra contains all subsets of null sets.
An outer measure for a nonempty set X is a function μ^{*}: 𝒫(X) → [0, ∞] that satisfies
(13)Let μ^{*} be an outer measure. A set A is μ^{*}measurable if for all E ⊆ X
(15)If 𝒜 ⊆ 𝒫(X) is an algebra, then μ_{0}: 𝒜 → [0, ∞] is a premeasure if
(16)Let F: ℝ → ℝ be an increasing, rightcontinuous function. The LebesgueStieltjes measure is the unique measure μ_{F} such that μ_{F}((a, b]) = F(b)  F(a).
The Lebesque measure is the LebesgueStieltjes measure for the function F(x) = x.
If (X, ℳ) and (Y, 𝒩) are measurable spaces, then f: X → Y is (ℳ, 𝒩)measurable if f^{1}(E) ∈ ℳ for all E ∈ 𝒩.
For any E ⊆ X, the characteristic function χ_{E} of E is defined as
(17)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)where E_{j} = f^{1}({z_j}) and range(f) = {z_{1}, …, z_{n}}.
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)The integral of φ is:
(20)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)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)The negative part of f is
(23)The integral of a real valued function f is defined as
(24)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)The space of complex valued functions f for which ∫_{E}f < ∞ is denoted L^{1}.
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 {f_{n}} defined on X converges uniformly to f if for every ε > 0 there exists N such that when n > N, f_{n}(x)  f(x) < ε for all x ∈ X.
A sequence of functions {f_{n}} defined on X converges pointwise to f if for every x ∈ X and every ε > 0 there exists N which may depend on x such that when n > N, f_{n}(x)  f(x) < ε.
A sequence of functions {f_{n}} 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 σalgebra 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)If (X, ℳ) and (Y, 𝒩) are measurable spaces and E ⊆ X × Y, the xsection E_{x} is the set {y ∈ Y: (x, y) ∈ E}.
Similarly if (X, ℳ) and (Y, 𝒩) are measurable spaces and E ⊆ X × Y, the ysection E^{y} is the set {x ∈ X: (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 {E_{j}} is a sequence of disjoint sets in ℳ, then
(27)where the latter sum converges absolutely if
(28)is finite.
A positive measure is a signed measure with range [0, ∞]. This is a synonym for a measure.
f is an extended μintegrable 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 F ⊆ E. If ν(F) ≤ 0 or ν(F) = 0 for all F ⊆ E 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 P ⋃ N = 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 E ⋃ F = 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 σfinite if its total variation is σfinite.
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
RadonNikodym derivative
TODO
 Lebesgue measure
 Lebesgue integral
 LebesgueStieltjes measure
 LebesgueStieltjes integral
 Riemann Integral
 types of convergence
 product σalgebras
 product measures
Theorems
Monotonicity
If (X, ℳ, μ) is a measure space, E, F ∈ ℳ, and E ⊆ F, then μ(E) ≤ μ(F).
Subadditivity
If (X, ℳ, μ) is a measure space and
(29)then
(30)Continuity from below
If (X, ℳ, μ) is a measure space,
(31)and E_{1} ⊆ E_{2} ⊆ ⋯, then
(32)Continuity from above
If (X, ℳ, μ) is a measure space,
(33)and E_{1} ⊇ E_{2} ⊇ ⋯, then
(34)Carathéodory's Theorem
If μ^{*} is an outer measure on X, the collection ℳ of μ^{*}measurable sets is a σalgebra, and the restriction of μ^{*} to ℳ is a complete measure.
Monotone Convergence
If {f_{n}} is a sequence in L^{+} such that f_{j} ≤ f_{j+1} for all j, and
(35)then
(36)Fatou's Lemma
If {f_{n}} is any sequence in L^{+}, then
(37)Dominated Convergence Theorem
Let {f_{n}} be a sequence in L^{1} such that (a) f_{n} → f a.e., and (b) there exists a nonnegative g ∈ L^{1} such that f_{n} ≤ g a.e. for all n. Then f ∈ L^{1} and ∫ f = lim_{n → ∞} ∫ f_{n}.
Egoroff's Theorem
Suppose that μ(X) < ∞ and f_{1}, f_{2}, … and f are measurable complexvalued functions on X and f_{n} → f a.e. Then for every ε > 0 there exists E ⊆ X such that μ(E) < ε and f_{n} → f uniformly on E^{C}.
Monotone Class Lemma
If 𝒜 is an algebra of subsets of X, then the monotone class 𝒞 generated by 𝒜 coincides with the σalgebra ℳ generated by 𝒜.
FubiniTonelli
Suppose that (X, ℳ, μ) and (Y, 𝒩, ν) are σfinite measure spaces.
a. (Tonelli) If f ∈ L^{+}(X × Y), then the functions g(x) = ∫ f_{x} dν and h(y) = ∫ f^{y} dμ are in L^{+}(X) and L^{+}(Y), respectively, and
(38)b. (Fubini) If f ∈ L^{1}(μ × ν), then f_{x} ∈ L^{1}(ν) for a.e. x ∈ X, f^{y} ∈ L^{1} for a.e. y ∈ Y, the a.e.defined functions g(x) = ∫ f_{x} dν and h(y) = ∫ f^{y}dμ are in L^{1}(μ) and L^{1}(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 P ⋃ N = X and P ⋂ N = ∅. 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^{}.
LebesgueRadonNikodym Theorem
Let ν be a σfinite signed measure and μ a σfinite positive measure on (X, ℳ). There exist unique σfinite signed measures λ, ρ on (X, ℳ) such that
(39)Moreover, there is an extended μintegrable function f: X → ℝ such that dρ = f dμ, and any two such functions are equal μa.e.