Monotone class theorem

Key takeaways

Monotone class theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Monotone class theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Monotone class theorem from memory before moving on to harder problems.

Reference excerpt

In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets G {\displaystyle G} is precisely the smallest 𝜎-algebra containing G . {\displaystyle G.} It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

Definition of a monotone class A monotone class is a family (i.e. class) M {\displaystyle M} of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means M {\displaystyle M} has the following properties:

if A 1 , A 2 , … ∈ M {\displaystyle A_{1},A_{2},\ldots \in M} and A 1 ⊆ A 2 ⊆ ⋯ {\displaystyle A_{1}\subseteq A_{2}\subseteq \cdots } then ⋃ i = 1 ∞ A i ∈ M , {\textstyle {\textstyle \bigcup \limits _{i=1}^{\infty }}A_{i}\in M,} and if B 1 , B 2 , … ∈ M {\displaystyle B_{1},B_{2},\ldots \in M} and B 1 ⊇ B 2 ⊇ ⋯ {\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots } then ⋂ i = 1 ∞ B i ∈ M . {\textstyle {\textstyle \bigcap \limits _{i=1}^{\infty }}B_{i}\in M.}

Monotone class theorem for sets

Monotone class theorem for functions

Proof The following argument originates in Rick Durrett's Probability: Theory and Examples.

Results and applications As a corollary, if G {\displaystyle G} is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of G . {\displaystyle G.}

By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a 𝜎-algebra. The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.

See also Dynkin system – Family closed under complements and countable disjoint unions π-𝜆 theorem – Family closed under complements and countable disjoint unionsPages displaying short descriptions of redirect targets π-system – Family of sets closed under intersection σ-algebra – Algebraic structure of set algebra

Citations

References Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.

Worked examples

Example 1 — a first encounter with Monotone class theorem

Start with the simplest possible case. Write down what Monotone class theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Monotone class theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Monotone class theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Monotone class theorem

In research: Monotone class theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Monotone class theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Monotone class theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Families of sets, Theorems in measure theory, so understanding it makes those chapters shorter.
In everyday life: Look for Monotone class theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Monotone class theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Monotone class theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Monotone class theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Monotone class theorem in simple terms?

In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets G {\displaystyle G} is precisely the smallest 𝜎-algebra containing G . {\displaystyle G.} It is used as a type of…

Why does Monotone class theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Monotone class theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Monotone class theorem.