Measure Theory: Complete Study Guide

1. Sigma-Algebras and Measurable Spaces

The starting point of measure theory is understanding which sets can be assigned a size. The naive hope that every subset of R can be consistently measured turns out to be false — the Axiom of Choice allows the construction of so-called non-measurable sets (such as the Vitali set). The solution is to restrict attention to a structured collection of sets called a sigma-algebra.

Definition: Sigma-Algebra

A sigma-algebra (also written as sigma-field) on a set X is a collection M of subsets of X satisfying three axioms:

From these axioms, one derives closure under countable intersections (by De Morgan's law), countable set differences, and finite operations. The pair (X, M) is called a measurable space, and sets in M are called measurable sets.

Examples of Sigma-Algebras

The two extreme examples are the trivial sigma-algebra (containing only the empty set and X itself) and the power set 2^X (containing every subset of X). Between these extremes lie the sigma-algebras of greatest practical importance.

2. Measures: Definitions and Basic Properties

A measure assigns a non-negative extended real number to each measurable set, generalizing notions of length, area, volume, counting, and probability.

Definition: Measure

The triple (X, M, mu) is a measure space. The countable additivity axiom (also called sigma-additivity) is far stronger than finite additivity and is what makes the theory powerful. Note that mu can take the value +infinity.

Fundamental Properties of Measures

Null Sets and Completeness

A null set (or set of measure zero) is a measurable set E with mu(E) = 0. Null sets can be large in cardinality: the Cantor set is uncountable yet has Lebesgue measure zero. A property is said to hold almost everywhere (a.e.) if the set where it fails is a null set.

A measure space (X, M, mu) is complete if every subset of a null set is measurable (and hence also a null set). The Lebesgue measure space is complete; the Borel measure space is not. Completing a measure space means adding all subsets of null sets to the sigma-algebra — this is the Lebesgue completion of the Borel sigma-algebra.

Examples of Measures

3. Lebesgue Measure on R^n

The Lebesgue measure is the canonical measure on Euclidean space. Its construction is non-trivial: one must prove the existence of a measure on the Borel (or Lebesgue) sigma-algebra that assigns to each rectangle its classical volume.

Construction via Outer Measure

The Lebesgue outer measure m^* of any subset E of R is defined by:

The outer measure is defined on all subsets of R (not just measurable ones), but it is only countably subadditive, not countably additive. The Caratheodory extension theorem identifies which sets can be measured consistently: a set E is Caratheodory-measurable if for every set A:

Caratheodory's theorem states that the Caratheodory-measurable sets form a sigma-algebra, and the restriction of m^* to this sigma-algebra is a complete measure. This sigma-algebra contains all Borel sets, and the resulting measure is Lebesgue measure m.

Key Properties of Lebesgue Measure

The Cantor Set: An Instructive Example

The Cantor set C is constructed by removing the middle third (1/3, 2/3) from [0,1], then the middle thirds of the remaining two intervals, and so on, countably many times. The total length removed is:

So m(C) = 1 - 1 = 0: the Cantor set has Lebesgue measure zero. Yet C is uncountable (it bijects with [0,1] via ternary expansions using digits 0 and 2), perfect (closed and every point is a limit point), and nowhere dense (its interior is empty). The Cantor set demonstrates that "small in measure" and "small in cardinality" are entirely different notions.

4. Measurable Functions

Just as continuity is the right notion of "nice" function for topology, measurability is the right notion for measure theory. Measurable functions are precisely those whose preimages of measurable sets are measurable.

Definition: Measurable Function

Stability Properties

The class of measurable functions is closed under all operations one might want to perform:

• •Sums, differences, products, quotients (when the denominator is non-zero a.e.) of measurable functions are measurable.
• •If f_n are measurable, then sup f_n, inf f_n, limsup f_n, and liminf f_n are measurable. In particular, pointwise limits of measurable functions are measurable.
• •Every continuous function from R to R is Borel measurable (measurable with respect to the Borel sigma-algebras).
• •Compositions: if f is measurable and g is continuous (or more generally Borel measurable), then g composed with f is measurable.

Simple Functions

A simple function is a measurable function taking only finitely many values. If the values are c_1, ..., c_n and the preimages are E_1, ..., E_n (which partition X), then the standard representation is:

Simple functions are the building blocks of Lebesgue integration. The key approximation theorem states: every non-negative measurable function f is the pointwise limit of an increasing sequence of non-negative simple functions. Moreover, if f is bounded, the convergence is uniform.

5. The Lebesgue Integral

The Lebesgue integral is constructed in three stages: first for non-negative simple functions, then for non-negative measurable functions via approximation, and finally for general measurable functions by splitting into positive and negative parts.

Stage 1: Integration of Simple Functions

For a non-negative simple function phi with standard representation phi = sum c_i times 1_(E_i), the integral is defined as the weighted sum of measures:

One checks this is well-defined (independent of the representation of phi) and that integration of simple functions is linear and monotone.

Stage 2: Integration of Non-Negative Functions

For a non-negative measurable function f, define:

This value in [0, +infinity] is always well-defined. The Monotone Convergence Theorem (proved at this stage) is crucial: if phi_n increases to f, then the integral of phi_n approaches the integral of f. This justifies the approximation procedure.

Stage 3: Integration of General Functions

For a general measurable function f, decompose it into positive and negative parts:

Then f is called integrable (or in L^1) if both integrals of f^+ and f^- are finite, and one defines:

Comparison with the Riemann Integral

If f is Riemann integrable on [a, b], then f is Lebesgue integrable and the two integrals agree. The converse fails: the Dirichlet function (1 on rationals, 0 on irrationals) has Lebesgue integral 0 but is not Riemann integrable. A bounded function on [a, b] is Riemann integrable if and only if it is continuous almost everywhere — this is the Lebesgue characterization of Riemann integrability.

6. The Three Great Convergence Theorems

The power of the Lebesgue integral over the Riemann integral lies in its convergence theorems, which justify interchanging limits and integrals under much weaker hypotheses.

Monotone Convergence Theorem (MCT)

Proof sketch: Since the sequence is increasing, its integrals form an increasing sequence of extended reals, converging to some limit L. Since f_n is at most f, we have L at most integral f d(mu). For the reverse inequality, fix epsilon in (0, 1) and a simple function phi with 0 at most phi at most f. Let A_n = (x : f_n(x) is at least epsilon times phi(x)). Then A_n increases to X, and by continuity from below: integral f_n d(mu) is at least integral over A_n of f_n d(mu) is at least epsilon times integral over A_n of phi d(mu), which approaches epsilon times integral phi d(mu). Since epsilon and phi were arbitrary, L is at least integral f d(mu).

Fatou's Lemma

Proof sketch: Apply MCT to g_n = inf_(k at least n) f_k, which increases to liminf f_n. Since g_n is at most f_k for all k at least n, we have integral g_n d(mu) is at most inf_(k at least n) integral f_k d(mu). Taking the limit and applying MCT gives the result.

Fatou's lemma shows the integral of the limit infimum is at most the limit infimum of the integrals. The inequality can be strict: consider f_n = n times 1_(0, 1/n) on [0, 1]. Each f_n has integral 1, but f_n converges to 0 pointwise, and the integral of 0 is 0.

Dominated Convergence Theorem (DCT)

Proof sketch: Apply Fatou's lemma to g + f_n (non-negative) and g minus f_n (non-negative) separately. Adding the resulting inequalities and using that integral g is finite yields the conclusion.

The DCT is perhaps the most-used tool in analysis. Typical applications include: differentiating under the integral sign (verify the derivative is bounded by an integrable function), justifying power series integration term by term, and proving continuity of parameter-dependent integrals.

7. L^p Spaces

L^p spaces are the natural function spaces arising in measure theory, providing the framework for harmonic analysis, PDEs, and functional analysis. They are Banach spaces — complete normed vector spaces — with rich structure.

Definitions

For a measure space (X, M, mu) and 1 at most p less than infinity, define:

Technically, L^p consists of equivalence classes of functions that agree almost everywhere, so that ||f||_p = 0 implies f = 0 (the positive definiteness of the norm requires this identification). When we write "f in L^p," we always mean an equivalence class.

Holder's Inequality

Proof sketch: Without loss of generality normalize so ||f||_p = ||g||_q = 1. Apply Young's inequality (ab at most a^p/p + b^q/q for non-negative a, b) to |f(x)| and |g(x)|, then integrate both sides.

The special case p = q = 2 is the Cauchy-Schwarz inequality for integrals. Holder's inequality is used to prove Minkowski's inequality, to establish duality of L^p spaces, and throughout PDEs to estimate norms.

Minkowski's Inequality (Triangle Inequality for L^p)

This is precisely the triangle inequality, confirming that ||.||_p is a genuine norm. Proof: write |f + g|^p at most |f| times |f + g|^(p-1) + |g| times |f + g|^(p-1) and apply Holder to each term with exponents p and q.

Completeness: The Riesz-Fischer Theorem

The most important structural theorem about L^p spaces:

Proof sketch for p less than infinity: Given a Cauchy sequence (f_n), extract a subsequence (f_n_k) with ||f_(n_(k+1)) minus f_(n_k)||_p less than 2^(-k). Let g = sum |f_(n_(k+1)) minus f_(n_k)|. By Minkowski and MCT, ||g||_p is finite, so g is finite a.e. The telescoping sum converges absolutely a.e. to some f in L^p, and one shows f_n converges to f in L^p norm.

L^2 as a Hilbert Space

The space L^2(mu) carries an inner product:

L^2 is a Hilbert space. This additional structure (orthogonality, projections, orthonormal bases) makes L^2 the central object in Fourier analysis, quantum mechanics, and the spectral theory of operators. The Fourier series of a function converges in L^2 norm by the completeness of the trigonometric system.

Inclusions and Relations Between L^p Spaces

On a finite measure space with mu(X) less than infinity: L^q is contained in L^p whenever p is at most q (larger exponents impose more integrability). On infinite measure spaces, no such inclusion holds in general. The dual space of L^p (for 1 less than p less than infinity) is L^q where 1/p + 1/q = 1. The dual of L^1 is L^infinity; the dual of L^infinity is strictly larger than L^1.

8. Product Measures and the Fubini-Tonelli Theorem

Product measures formalize the notion of multi-dimensional integration and justify switching the order of iterated integrals.

Construction of the Product Measure

Given two sigma-finite measure spaces (X, M, mu) and (Y, N, nu), the product sigma-algebra M times-cross N is generated by measurable rectangles A times B (with A in M, B in N). The product measure mu times nu is the unique measure on M times-cross N satisfying:

Existence and uniqueness follow from the Caratheodory extension theorem applied to the premeasure defined on rectangles. The sigma-finiteness hypothesis is necessary for uniqueness.

Tonelli's Theorem (Non-Negative Functions)

Fubini's Theorem (Integrable Functions)

The standard workflow is: apply Tonelli to |f| to verify integrability (if one iterated integral of |f| is finite, then f is in L^1), then apply Fubini to f itself to switch the order of integration freely.

9. Signed Measures and the Radon-Nikodym Theorem

Signed measures generalize measures by allowing negative values. They arise naturally as differences of measures and as indefinite integrals of integrable functions that change sign.

Signed Measures

A signed measure nu on (X, M) is a function nu: M to [-infinity, +infinity] satisfying nu(empty set) = 0 and countable additivity for pairwise disjoint sequences (with the constraint that at most one of +infinity, -infinity is attained, and the sum converges absolutely when both finite and infinite terms are present).

Hahn Decomposition Theorem

From the Hahn decomposition, one obtains the Jordan decomposition nu = nu^+ minus nu^- where nu^+(E) = nu(E intersect P) and nu^-(E) = -nu(E intersect N) are both positive measures. The total variation measure is |nu| = nu^+ + nu^-.

Absolute Continuity and Singularity

Two measures mu and nu on the same space can be related in two extreme ways:

Lebesgue-Radon-Nikodym Theorem

The Radon-Nikodym theorem is the measure-theoretic analog of the Fundamental Theorem of Calculus. In probability theory, if P and Q are probability measures with P absolutely continuous with respect to Q, then dP/dQ is called the likelihood ratio or density, central to Bayesian inference and change-of-measure arguments (the Girsanov theorem in stochastic calculus).

10. Modes of Convergence

In analysis, "convergence" can mean many different things. Understanding the relationships between modes of convergence is essential for applying the right theorem.

The Four Main Modes

Implications and Counterexamples

The relationships between modes of convergence, and the failure of converses, are captured in the following diagram and counterexamples:

11. Egorov's Theorem and Lusin's Theorem

Two classical theorems connect measure-theoretic and topological notions of "almost uniform" behavior of measurable functions.

Egorov's Theorem

In words: a.e. convergence on a finite measure space is "almost uniform" — you can excise a set of arbitrarily small measure and get uniform convergence on the rest.

Proof sketch: Define A_(n, k) = (x : |f_m(x) minus f(x)| at least 1/k for some m at least n). For fixed k, A_(n, k) decreases to the null set (x : f_m(x) does not converge to f(x)) as n increases, so by continuity from above mu(A_(n, k)) to 0. Choose n_k large enough that mu(A_(n_k, k)) less than epsilon/2^k. Let F = union A_(n_k, k). Then mu(F) less than epsilon and convergence on F^c is uniform.

The finiteness hypothesis is necessary: on R with Lebesgue measure, the sequence f_n = 1_(n, n+1) converges to 0 everywhere but not almost uniformly (any set of finite measure misses all but finitely many of the bumps).

Lusin's Theorem

This is sometimes paraphrased as: "every measurable function is nearly continuous" — continuous on all but an arbitrarily small set. Note that f is NOT being claimed to be continuous; only its restriction to K is continuous.

Lusin's theorem implies that every measurable function can be approximated in L^p by continuous functions of compact support, which is used to prove density of smooth functions in L^p spaces.

12. Probability as Measure Theory

Modern probability theory is built entirely on measure theory. The measure-theoretic framework unifies discrete, continuous, and mixed probability distributions and provides the tools to prove limit theorems rigorously.

The Kolmogorov Probability Space

A probability space is a measure space (Omega, F, P) where:

This framework, introduced by Kolmogorov in 1933, resolved longstanding foundational questions and unified all of probability theory under a single mathematical structure.

Random Variables as Measurable Functions

A random variable X is a measurable function X: (Omega, F) to (R, B(R)). The law (or distribution) of X is the pushforward measure P_X = P composed with X^(-1) on B(R):

Two random variables with the same law are identically distributed even if they live on entirely different probability spaces. This abstraction allows one to work with distributions without specifying the underlying probability space.

Expectation as Lebesgue Integral

The expectation of a random variable X is its Lebesgue integral with respect to P:

For discrete X: E[X] = sum x_i P(X = x_i) (the familiar formula). For absolutely continuous X with density f_X: E[X] = integral x f_X(x) dx. Both are special cases of the Lebesgue integral. The DCT becomes the Dominated Convergence Theorem for expectations: if X_n to X a.s. and |X_n| is at most Y with E[Y] finite, then E[X_n] to E[X].

Independence via Product Measures

Events A and B are independent if P(A intersect B) = P(A) times P(B). Random variables X and Y are independent if the sigma-algebras they generate are independent, which is equivalent to: the joint law P_(X,Y) equals the product measure P_X times P_Y on B(R^2). This product measure perspective immediately gives: E[XY] = E[X] times E[Y] for independent X, Y (by Fubini).

Strong Law of Large Numbers

The "almost surely" is precisely the measure-theoretic a.e.: the set of outcomes omega for which the averages do not converge to mu has probability zero. The proof uses:

• 1.Borel-Cantelli lemma (a measure-theoretic result about limsups of events): if sum P(A_n) is finite, then P(limsup A_n) = 0.
• 2.Kolmogorov's maximal inequality and truncation arguments to reduce to the case of bounded variables.
• 3.The MCT or DCT to handle the truncation error.

Conditional Expectation via Radon-Nikodym

The measure-theoretic definition of conditional expectation E[X | G] for a sub-sigma-algebra G of F is the unique G-measurable random variable Y satisfying:

Existence follows from the Radon-Nikodym theorem: the map A to integral over A of X dP defines a measure on (Omega, G) that is absolutely continuous with respect to P restricted to G, and Y is its Radon-Nikodym derivative. This abstract definition unifies conditioning on events, conditioning on discrete random variables, and conditioning on continuous random variables.

13. Frequently Asked Questions

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