Weakly measurable function

Key takeaways

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

Reference excerpt

In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree.

Definition If ( X , Σ ) {\displaystyle (X,\Sigma )} is a measurable space and B {\displaystyle B} is a Banach space over a field K {\displaystyle \mathbb {K} } (which is the real numbers R {\displaystyle \mathbb {R} } or complex numbers C {\displaystyle \mathbb {C} } ), then f : X → B {\displaystyle f:X\to B} is said to be weakly measurable if, for every continuous linear functional g : B → K , {\displaystyle g:B\to \mathbb {K} ,} the function

g ∘ f : X → K defined by x ↦ g ( f ( x ) ) {\displaystyle g\circ f\colon X\to \mathbb {K} \quad {\text{ defined by }}\quad x\mapsto g(f(x))}

is a measurable function with respect to Σ {\displaystyle \Sigma } and the usual Borel σ {\displaystyle \sigma } -algebra on K . {\displaystyle \mathbb {K} .}

A measurable function on a probability space is usually referred to as a random variable (or random vector if it takes values in a vector space such as the Banach space B {\displaystyle B} ). Thus, as a special case of the above definition, if ( Ω , P ) {\displaystyle (\Omega ,{\mathcal {P}})} is a probability space, then a function Z : Ω → B {\displaystyle Z:\Omega \to B} is called a ( B {\displaystyle B} -valued) weak random variable (or weak random vector) if, for every continuous linear functional g : B → K , {\displaystyle g:B\to \mathbb {K} ,} the function

g ∘ Z : Ω → K defined by ω ↦ g ( Z ( ω ) ) {\displaystyle g\circ Z\colon \Omega \to \mathbb {K} \quad {\text{ defined by }}\quad \omega \mapsto g(Z(\omega ))}

is a K {\displaystyle \mathbb {K} } -valued random variable (i.e. measurable function) in the usual sense, with respect to Σ {\displaystyle \Sigma } and the usual Borel σ {\displaystyle \sigma } -algebra on K . {\displaystyle \mathbb {K} .}

Properties The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem. A function f {\displaystyle f} is said to be almost surely separably valued (or essentially separably valued) if there exists a subset N ⊆ X {\displaystyle N\subseteq X} with μ ( N ) = 0 {\displaystyle \mu (N)=0} such that f ( X ∖ N ) ⊆ B {\displaystyle f(X\setminus N)\subseteq B} is separable.

In the case that B {\displaystyle B} is separable, since any subset of a separable Banach space is itself separable, one can take N {\displaystyle N} above to be empty, and it follows that the notions of weak and strong measurability agree when B {\displaystyle B} is separable.

See also Bochner measurable function Bochner integral – Concept in mathematics Bochner space – Type of topological space Pettis integral Vector measure – Generalization of finite measure to Banach spaces

References

Pettis, B. J. (1938). "On integration in vector spaces". Trans. Amer. Math. Soc. 44 (2): 277–304. doi:10.2307/1989973. ISSN 0002-9947. MR 1501970. Showalter, Ralph E. (1997). "Theorem III.1.1". Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. 103. ISBN 0-8218-0500-2. MR 1422252.

Worked examples

Example 1 — a first encounter with Weakly measurable function

Start with the simplest possible case. Write down what Weakly measurable function 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 Weakly measurable function 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 Weakly measurable function 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 Weakly measurable function

In research: Weakly measurable function 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 Weakly measurable function 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: Weakly measurable function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functional analysis, Measure theory, Types of functions, so understanding it makes those chapters shorter.
In everyday life: Look for Weakly measurable function 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 Weakly measurable function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Weakly measurable function 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 Weakly measurable function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Weakly measurable function in simple terms?

In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurabilit…

Why does Weakly measurable function 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 Weakly measurable function?

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 Weakly measurable function.