measurable function on

USAGE SUMMARY

The phrase "measurable function on" is correct and usable in written English.
It is typically used in mathematical contexts, particularly in discussions about functions that can be integrated or analyzed within a certain measurable space. Example: "Let f be a measurable function on the interval [a, b]."

✓ Grammatically correct

Science

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

The nonadditive set function μ is a mapping from P X) to [0, ∞) where μ=0. Let f: X→ be a measurable function on the measure space (X, F, μ), whose outputs represent the observed target values.

BMC Systems Biology

for a nonnegative Borel measurable function on and a nonnegative measure on a Borel set.

Journal of Inequalities and Applications

Let f be a measurable function on R n.

Journal of Inequalities and Applications

where f is a nonnegative measurable function on R n.

Journal of Inequalities and Applications

Let and let be a nonnegative measurable function on.

Journal of Inequalities and Applications

Let g be a measurable function on (mathbb {R}^{n}).

Journal of Inequalities and Applications

If u is a measurable function on ∂ C n satisfying.

Boundary Value Problems

Let f be a measurable function on (Bbb {R}^{n}).

Journal of Inequalities and Applications

Let φ ( x, r ) be a positive measurable function on R n × R + and w be non-negative measurable function on R n.

Journal of Inequalities and Applications

Suppose that is a measurable function on, and we have (see [10]) (2.17).

Journal of Inequalities and Applications

Expert writing Tips

Best practice

When using "measurable function on", clearly define the measure space on which the function is defined. This provides essential context for understanding the function's properties.

Common error

Avoid assuming that a "measurable function on" is necessarily continuous. Measurability is a weaker condition than continuity; a function can be measurable without being continuous.

Linguistic Context

The phrase "measurable function on" serves as a mathematical descriptor, specifying a function that adheres to the conditions of measurability within a defined space. This is confirmed by Ludwig AI, which validates the phrase's correctness and usability in mathematical contexts.

Expression frequency: Very common

Frequent in

Science

100%

Less common in

News & Media

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Formal & Business

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Academia

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Ludwig's WRAP-UP

The phrase "measurable function on" is a core concept in mathematical analysis, particularly in measure theory and integration. As confirmed by Ludwig AI, this phrase is grammatically sound and frequently used in scientific and academic contexts. It's crucial to define the measure space when using this phrase to provide context. Common errors include assuming that measurability implies continuity, a concept to be carefully avoided. While alternatives exist, they often specify stronger conditions (e.g., continuous, differentiable). The phrase is most commonly found in science-related sources, reflecting its specialized usage. Understanding its function, purpose, and register is vital for accurate and effective communication in mathematical discourse.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

Lebesgue measurable function on

This specifies a function measurable with respect to the Lebesgue measure, providing a context for integration.

Borel measurable function on

This indicates a function measurable with respect to a Borel sigma-algebra, a specific type of measurability.

non-negative function on

This indicates a function whose values are always greater than or equal to zero, restricting the sign of the output.

real-valued function on

This specifies that the function's output values are real numbers, focusing on the nature of the codomain.

complex-valued function on

This specifies that the function's output values are complex numbers, expanding the range beyond real numbers.

integrable function on

This term refers to a function whose integral exists, implying a stronger condition than measurability.

square-integrable function on

This represents a function whose square has a finite integral, important in functional analysis.

differentiable function on

This refers to a function with a derivative at each point in its domain, indicating smoothness.

continuous function on

This describes a function without any abrupt interruptions or breaks in its graph, a fundamental concept in calculus.

bounded function on

This indicates a function whose values are constrained within a specific range, restricting its magnitude.

FAQs

How is a "measurable function on" used in mathematics?

In mathematics, a "measurable function on" a set is a function for which the inverse image of any measurable set is also measurable. This is a fundamental concept in measure theory and is crucial for defining integration.

What are some examples of "measurable function on" used in science?

In various scientific domains, a "measurable function on" can represent physical quantities, such as temperature distribution on a surface or probability densities in statistical mechanics. These functions allow for quantitative analysis and predictions.

Which properties does a "measurable function on" need to satisfy?

A "measurable function on" needs to satisfy the condition that for any measurable set in the codomain, its preimage in the domain is also measurable. This ensures that the function preserves the structure of measurable sets under the mapping.

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

A continuous function preserves topological properties, whereas a "measurable function on" preserves measure-theoretic properties. Continuity implies measurability, but the converse is not always true. Many functions can be measurable but not continuous.