Basics of Measure Theory
Y. Eddie Lu, Summer 2026
ECON 8002 course index · Lecture 1 of 9
Orientation
How to use this page
Answer each quiz before opening it. For every theorem, name the objects, then the assumption, then the conclusion. Do not confuse an event, its indicator, and its probability.
How should I think about this lecture?
Which subsets are legitimate events?
How can a numerical variable be defined from a state?
\(\longrightarrow\)
Probability = a measure on observable events; random variables = measurable maps
\(\longrightarrow\)
CDFs and joint laws
Expectation and integration next lecture
Course map
- Measure spaces (current lecture)
- Integration and Radon–Nikodym derivatives
- Properties of probability distributions
- Conditional expectation and independence
- Convergence modes and stochastic orders
- Continuous mapping and Slutsky’s theorem
- Laws of large numbers
- Weak convergence and the delta method
- Central limit theorems and inference
Today supplies the sample spaces, events, laws, and random variables used everywhere else.
Prerequisite retrieval
Let \(A,B\subseteq\Omega\). Recall \(A\setminus B=A\cap B^c\) and \((\bigcup_n A_n)^c=\bigcap_n A_n^c\). A function \(f:\Omega\to S\) sends a state to a value; \(f^{-1}(C)=\{\omega:f(\omega)\in C\}\) sends a set of values back to states.
Retrieval check.
If \(f(x)=x^2\), what is \(f^{-1}([1,4])\)?
\([-2,-1]\cup[1,2]\). This is a preimage, so \(f\) need not have an inverse function.
Sigma-algebras and measurable spaces
Motivation: a two-state asset payoff
Let \(\Omega=\{u,d\}\) denote tomorrow’s market state. An event is, for example, \(A=\{u\}\). A payoff \(X\) might satisfy \(X(u)=1.10\) and \(X(d)=0.90\). Probability assigns numbers to events, while \(X\) maps states to numbers.
The triple \((\Omega,\mathcal F,P)\) says what can happen, which questions about it are meaningful, and how likely those questions are.
Why not use every subset?
For finite \(\Omega\), we usually can take \(\mathcal F=2^\Omega\). For uncountable spaces such as \(\mathbb R\), requiring probabilities for every subset conflicts with desirable translation-invariant length properties. We therefore specify a tractable collection of measurable subsets.
Definition: \(\sigma\)-algebra
A collection \(\mathcal F\subseteq2^\Omega\) is a \(\sigma\)-algebra if:
- \(\varnothing\in\mathcal F\);
- \(A\in\mathcal F\Rightarrow A^c\in\mathcal F\);
- \(A_1,A_2,\ldots\in\mathcal F\Rightarrow\bigcup_{n=1}^\infty A_n\in\mathcal F\).
Consequences of the definition
Because \(\varnothing\in\mathcal F\), also \(\Omega=\varnothing^c\in\mathcal F\). By De Morgan, \[\bigcap_{n=1}^\infty A_n=\left(\bigcup_{n=1}^\infty A_n^c\right)^c\in\mathcal F.\] Thus \(\mathcal F\) is closed under countable intersections and finite unions/intersections.
Example and nonexample
For \(A\subseteq\Omega\), \(\{\varnothing,A,A^c,\Omega\}\) is a \(\sigma\)-algebra. By contrast, \(\{\varnothing,\{1\},\{2\}\}\) on \(\Omega=\{1,2\}\) is not: the countable union \(\{1\}\cup\{2\}=\Omega\) is missing.
Retrieval check.
Why is closure under countable unions stronger than closure under finite unions?
Every finite union is a countable union: for \(A_1\cup\cdots\cup A_m\), set \(A_{m+1}=A_{m+2}=\cdots=\varnothing\). Closure under countable unions therefore includes closure under finite unions and additionally permits countably infinite unions. The converse need not hold.
Measurable spaces and generated information
\((\Omega,\mathcal F)\) is a measurable space. Elements of \(\mathcal F\) are measurable sets or events. For any collection \(\mathcal C\subseteq2^\Omega\), \(\sigma(\mathcal C)\) is the smallest \(\sigma\)-algebra containing \(\mathcal C\).
The smallest and largest choices are \(\{\varnothing,\Omega\}\) and \(2^\Omega\).
Borel sets
On \(\mathbb R\), the Borel \(\sigma\)-algebra is \[\mathcal B(\mathbb R)=\sigma\bigl(\{(a,b):a<b\}\bigr).\] It contains open and closed sets and all usual intervals. On \(\mathbb R^d\), use \(\mathcal B(\mathbb R^d)\).
Do not say “all subsets of \(\mathbb R\) are Borel.” They are not.
A Vitali set \(V\subseteq[0,1]\) is formed by selecting one representative from each equivalence class modulo \(\mathbb Q\). It is not Lebesgue measurable. Every Borel set is Lebesgue measurable, so \(V\) is not Borel.
Measures and distribution functions
Definition: measure
On \((\Omega,\mathcal F)\), a measure is a map \(\mu:\mathcal F\to[0,\infty]\) satisfying \(\mu(\varnothing)=0\) and, for pairwise disjoint \(A_n\in\mathcal F\), \[\mu\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty\mu(A_n).\] A probability measure has \(P(\Omega)=1\).
Countable additivity is the central restriction
Disjoint pieces do not double-count. The range includes \(\infty\), so counting measure on an infinite set is legitimate. But \(\infty-\infty\) and \(\infty/\infty\) remain undefined.
Three measures
| Measure | Space | Meaning |
|---|---|---|
| Counting | \((\mathbb N,2^{\mathbb N})\) | \(\#A\) |
| Lebesgue \(\lambda\) | \((\mathbb R,\mathcal B)\) | interval length |
| Dirac \(\delta_{x_0}\) | \((\mathbb R,\mathcal B)\) | \(\delta_{x_0}(A)=\mathbf1\{x_0\in A\}\) |
Retrieval check.
Can normalized counting measure give a uniform probability distribution on all of \(\mathbb N\)?
No. Each singleton would need equal mass; positive mass sums to \(\infty\), while zero mass sums to \(0\).
Proposition 1.1: three workhorses
For a measure \(\mu\):
- \(A\subseteq B\Rightarrow\mu(A)\leq\mu(B)\).
- \(\mu(\bigcup_nA_n)\leq\sum_n\mu(A_n)\).
- If \(A_n\uparrow A\), then \(\mu(A_n)\uparrow\mu(A)\).
Proof map: monotonicity and subadditivity
- Write \(B=A\sqcup(B\setminus A)\), where \(\sqcup\) denotes a disjoint union.
- Apply additivity and nonnegativity to get \(\mu(B)\geq\mu(A)\).
- For arbitrary \(A_n\), disjointify: \(C_n=A_n\setminus\bigcup_{j<n}A_j\).
- Then \(\bigcup_nC_n=\bigcup_nA_n\) and \(\mu(C_n)\leq\mu(A_n)\).
Proof map: continuity from below
Set \(C_1=A_1\) and \(C_n=A_n\setminus A_{n-1}\). The \(C_n\) are disjoint and \(A_n=\bigcup_{j\le n}C_j\), so countable additivity gives \[\mu\left(\bigcup_nA_n\right)=\sum_n\mu(C_n)=\lim_n\mu(A_n).\]
Continuity from above
If \(A_n\downarrow A\) and \(\mu(A_1)<\infty\), then \[\mu(A_n)\downarrow\mu(A).\]
Apply continuity from below to \(A_1\setminus A_n\uparrow A_1\setminus A\), then subtract from the finite number \(\mu(A_1)\). The finiteness condition prevents an undefined \(\infty-\infty\) argument.
Retrieval check.
Which assumption is missing from \(\mu(A\cup B)=\mu(A)+\mu(B)\)?
Disjointness. Otherwise the intersection is counted twice.
CDF: a probability measure summarized on half-lines
For a probability measure \(P\) on \((\mathbb R,\mathcal B)\), \[F(x)=P(( -\infty,x])\] is its cumulative distribution function.
CDF characterization
A CDF is nondecreasing and right-continuous, and \[\lim_{x\to-\infty}F(x)=0,\qquad \lim_{x\to+\infty}F(x)=1.\] Conversely, any function with these four properties is the CDF of a unique probability measure on \((\mathbb R,\mathcal B)\).
Right continuity follows from measuring decreasing half-lines \(( -\infty,x+1/n] \downarrow( -\infty,x]\).
Retrieval check.
Can a CDF jump? What does a jump at \(x\) equal?
Yes. The jump is \(P(X=x)\), an atom of the distribution.
Product spaces and joint laws
Product \(\sigma\)-algebras
For measurable spaces \((\Omega_j,\mathcal F_j)\), the product \(\sigma\)-algebra is \[\bigotimes_{j=1}^d\mathcal F_j =\sigma\{A_1\times\cdots\times A_d:A_j\in\mathcal F_j\}\] on \(\prod_{j=1}^d\Omega_j\). The Cartesian product of the collections \(\mathcal F_j\) is not itself generally a \(\sigma\)-algebra.
\(\sigma\)-finiteness and product measures
A measure \(\mu\) is \(\sigma\)-finite if there are \(E_1,E_2,\ldots\in\mathcal F\) with \(\Omega=\bigcup_nE_n\) and \(\mu(E_n)<\infty\) for every \(n\).
If each \(\mu_j\) is \(\sigma\)-finite, there exists a unique product measure \(\mu_1\otimes\cdots\otimes\mu_d\) on \(\bigotimes_j\mathcal F_j\) satisfying \[ (\mu_1\otimes\cdots\otimes\mu_d)(A_1\times\cdots\times A_d)=\prod_{j=1}^d\mu_j(A_j).\]
Joint and marginal CDFs
For \(X=(X_1,\ldots,X_d)\), \[F_X(x_1,\ldots,x_d)=P(X_1\le x_1,\ldots,X_d\le x_d).\] The \(j\)th marginal follows by sending every other coordinate to \(+\infty\).
Marginals do not determine the joint law. \(Y=X\) and \(Y=-X\) can both have standard-normal marginals.
Retrieval check.
If \(X,Y\sim N(0,1)\) marginally, does that determine \(P(XY>0)\)?
No. Dependence, hence the joint distribution, is still unspecified.
Measurable functions and probability laws
Inverse images
For \(f:\Omega\to S\) and \(B\subseteq S\), \[f^{-1}(B)=\{\omega\in\Omega:f(\omega)\in B\}.\] It obeys \(f^{-1}(B^c)=[f^{-1}(B)]^c\) and \(f^{-1}(\bigcup_nB_n)=\bigcup_nf^{-1}(B_n)\).
Measurable maps
A map \(f:(\Omega,\mathcal F)\to(S,\mathcal G)\) is measurable if \(f^{-1}(B)\in\mathcal F\) for every \(B\in\mathcal G\). For \(S=\mathbb R\) and \(\mathcal G=\mathcal B(\mathbb R)\), \(f\) is Borel measurable.
Random variables and information
On \((\Omega,\mathcal F,P)\), a measurable map \(X:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B)\) is a random variable. A measurable map into \(\mathbb R^d\) is a random vector. The collection \[\sigma(X)=\{X^{-1}(B):B\in\mathcal B(\mathbb R)\}\] is the information revealed by \(X\).
Retrieval check.
If \(A\in\mathcal F\), why is \(\mathbf1_A\) a random variable?
Preimages of Borel subsets of \(\{0,1\}\) are among \(\varnothing,A,A^c,\Omega\), all in \(\mathcal F\).
Closure: how measurable functions are built
Let \(f,g:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B)\) be finite-valued and measurable, and let \(a,b\in\mathbb R\). Then \(af+bg\) and \(fg\) are measurable; \(f/g\) is measurable on the measurable set \(\{g\ne0\}\). If \(f_n:\Omega\to\overline{\mathbb R}\) are measurable, then \(\sup_nf_n\), \(\inf_nf_n\), \(\limsup_nf_n\), and \(\liminf_nf_n\) are extended-real measurable. The set \(A=\{\omega:\lim_nf_n(\omega)\text{ exists in }\mathbb R\}\) is measurable, and the real-valued map equal to \(\lim_nf_n\) on \(A\) and \(f_1\) on \(A^c\) is measurable.
If \(f:(\Omega,\mathcal F)\to(S,\mathcal G)\) and \(h:(S,\mathcal G)\to(T,\mathcal H)\) are measurable, then \(h\circ f:(\Omega,\mathcal F)\to(T,\mathcal H)\) is measurable. A continuous map between Borel subsets of Euclidean spaces is Borel measurable.
Indicators and simple functions
For \(A\in\mathcal F\), an indicator is \(\mathbf1_A(\omega)=1\) on \(A\) and \(0\) elsewhere. A measurable simple function has finite range and a representation \[s(\omega)=\sum_{j=1}^m a_j\mathbf1_{A_j}(\omega).\] with every \(A_j\in\mathcal F\). Nonnegative measurable functions can be approximated from below by increasing nonnegative simple functions. This is the construction of the Lebesgue integral.
Retrieval check.
What is \(\sigma(\mathbf1_A)\) when \(A\notin\{\varnothing,\Omega\}\)?
\(\{\varnothing,A,A^c,\Omega\}\).
Distribution or law
The law of a random element \(X:(\Omega,\mathcal F,P)\to(S,\mathcal G)\) is the pushforward measure \[P_X(B)=P(X\in B)=P(X^{-1}(B)),\qquad B\in\mathcal G.\]
The law lives on the value space. It lets us study a return, estimator, or payoff without retaining the entire latent state space.
Synthesis and retrieval
Economic bridge
An estimator \(\widehat\theta_n\) is a measurable function of the sample. A statement such as \(\widehat\theta_n\xrightarrow{p}\theta_0\) is therefore a claim about the sequence of induced laws and probabilities on the sample space. Joint laws, not marginal summaries alone, govern portfolio risk and multivariate asymptotics.
Assumption audit
| Assumption | Job | Failure without it |
|---|---|---|
| \(\mathcal F\) is a \(\sigma\)-algebra | events survive countable operations | probabilities of limit events need not exist |
| countable additivity | links finite and infinite partitions | CDF/limit arguments break |
| measurability of \(X\) | \(\{X\in B\}\) is an event | law of \(X\) is undefined |
| \(\sigma\)-finiteness | product/RN theorems | standard existence results can fail |
Proof blueprints
| Claim | Reusable proof pattern |
|---|---|
| measure inequality | decompose into disjoint pieces |
| preimage identity | fix \(\omega\) and use iff statements |
| generated information | verify the three \(\sigma\)-algebra axioms |
| CDF property | translate half-lines into nested events |
Common errors
Do not write \(P(x)\) for a continuously valued realization. Do not call a preimage an inverse. Do not infer a joint law from marginals. Do not treat every function as a random variable without checking measurability.
Exit ticket 1
Retrieval check.
State the three axioms of a \(\sigma\)-algebra.
It contains \(\varnothing\), is closed under complements, and under countable unions.
Exit ticket 2
Retrieval check.
Prove \(A\subseteq B\Rightarrow\mu(A)\leq\mu(B)\).
\(B=A\sqcup(B\setminus A)\), so \(\mu(B)=\mu(A)+\mu(B\setminus A)\ge\mu(A)\).
Exit ticket 3
Retrieval check.
For a return \(R\), explain the difference between \(R\), \(\{R\le0\}\), and \(P_R\).
\(R\) is a measurable map, \(\{R\le0\}\) is an event in \(\mathcal F\), and \(P_R\) is the probability measure on return values induced by \(R\).
Mastery checklist
You should now be able to define a measure space, probability space, Borel random variable, law, and joint CDF; prove the basic measure inequalities; calculate preimages; and explain why a joint law contains information absent from its marginals.
Built from ECON 8002 Lecture 1, with corrected notation and expanded examples. The source labels a few duplicated proposition parts; this page uses the standard statements.