measurable function

USAGE SUMMARY

The phrase "measurable function" is correct and usable in written English.
It is typically used in mathematics and statistics to refer to a function that is compatible with a given measure, allowing for the integration and analysis of the function's properties. Example: "In probability theory, a random variable is defined as a measurable function from a sample space to the real numbers."

✓ Grammatically correct

Science

Encyclopedias

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

Suppose that f is a measurable function on a measure space.

Journal of Inequalities and Applications

Let ((Omega, Sigma, mu)) be a measure space, (f:Omegarightarrow [0,1]) be a measurable function.

Journal of Inequalities and Applications

Such a description has two constituents, the measurable function or random variable, and the probability measure.

Engineering Structures

The appropriate restriction is that a random variable must be a measurable function.

Encyclopedia Britannica

where is a measurable function.

Journal of Inequalities and Applications

(b) defines a measurable function from into.

Journal of Inequalities and Applications

The obstacle is a measurable function.

Boundary Value Problems

Let be a non (Lebesgue) measurable function.

Journal of Inequalities and Applications

Let (f:{mathbb{R}}^rightarrow X) be a measurable function.

Advances in Difference Equations

Let f be a measurable function on R n.

Journal of Inequalities and Applications

Let ω be a non-negative measurable function.

Journal of Inequalities and Applications

Expert writing Tips

Best practice

When using "measurable function", ensure that the context clearly defines the measure space and sigma-algebra being considered, as measurability is relative to these structures.

Common error

Do not assume that a "measurable function" is necessarily continuous. Measurability is a weaker condition than continuity, and many "measurable functions" can be discontinuous.

Linguistic Context

The phrase "measurable function" functions as a noun phrase, where "measurable" is an adjective modifying the noun "function". It typically denotes a function that satisfies the criteria of measurability with respect to a given sigma-algebra. Ludwig confirms the usage across various mathematical contexts.

Expression frequency: Very common

Frequent in

Science

75%

Encyclopedias

15%

Formal & Business

10%

Less common in

News & Media

0%

Social Media

0%

Reference

0%

Ludwig's WRAP-UP

The term "measurable function" is a crucial concept in mathematics, particularly in real analysis and probability theory. As Ludwig confirms, it describes a function that respects a given measure, allowing for integration and analysis. The phrase is predominantly used in formal and scientific contexts. While grammatically sound, it's essential to understand the context and measure space to ensure accurate usage. Understanding the distinction between measurability and continuity is also crucial to avoid common errors. "Measurable function" is very common in scientific literature, authoritative publications, and encyclopedias and less common in news media and social media.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

measurable mapping

Uses the term "mapping" instead of "function", emphasizing the transformation aspect.

real-valued measurable function

Emphasizes that the function's output values are real numbers.

complex-valued measurable function

Specifies that the function's output values are complex numbers.

Lebesgue measurable function

Explicitly states the function is measurable with respect to the Lebesgue measure, clarifying the measure being used.

integrable function

Focuses on the function's integrability rather than its measurability, implying it can be integrated.

square-integrable function

Highlights that the square of the function is integrable.

Borel function

Specifies a function measurable with respect to the Borel sigma-algebra, a more specific type of measurable function.

weakly measurable function

Describes a function where measurability applies to its composition with linear functionals.

random variable

In probability theory, a random variable is a measurable function from a sample space to the real numbers.

essentially bounded function

Indicates that the function is bounded almost everywhere.

FAQs

How is a "measurable function" used in probability theory?

In probability theory, a "random variable" is defined as a "measurable function" from a sample space (with a probability measure) to the real numbers. This allows us to apply measure theory to analyze probabilities of events.

What's the difference between a "measurable function" and a continuous function?

A continuous function preserves topological structure, whereas a "measurable function" preserves measurable structure. Every continuous function is measurable, but the converse isn't necessarily true. A "measurable function" can have discontinuities and still be measurable.

What does it mean for a function to be Lebesgue measurable?

A Lebesgue "measurable function" is a function where the preimage of every Borel set is Lebesgue measurable. This is a specific type of measurability with respect to the Lebesgue measure, commonly used in real analysis.

What are some examples of "measurable functions"?

Examples include continuous functions, step functions, and indicator functions. In general, any function whose level sets are measurable is a "measurable function". A "Borel function" is also a measurable function.