everywhere integrable (english) - meaning, definition, translation, pronunciation

Part of Speech

Adjective

Phonetic Transcription

/ˈɛvriˌwɛr ˈɪntəˌɡreɪbəl/

Meaning and Usage

The term "everywhere integrable" refers to a function that is integrable over the entire domain in the context of mathematics, specifically in measure theory and real analysis. A function is said to be everywhere integrable if its integral converges (i.e., it has a finite value) over its entire range. This concept is crucial when discussing properties of functions in calculus, especially in the context of Lebesgue integration.

This phrase is primarily used in written mathematical contexts, such as textbooks, academic papers, and lectures. It is not commonly found in everyday speech.

Example Sentences

1. A function that is everywhere integrable can be analyzed using various techniques in measure theory.

2. Translation: Функция, которая является интегрируемой повсюду, может быть проанализирована с использованием различных методов в теории меры.

3. In our analysis, we focus on identifying whether explicit examples of functions are indeed everywhere integrable.

4. Translation: В нашем анализе мы сосредотачиваемся на том, чтобы определить, являются ли конкретные примеры функций действительно интегрируемыми повсюду.

5. The importance of everywhere integrable functions lies in their applicability in solving real-world problems.

6. Translation: Важность интегрируемых повсюду функций заключается в их применимости при решении реальных задач.

Idiomatic Expressions

The term "everywhere integrable" does not have widely recognized idiomatic expressions associated with it, as it is highly specific to mathematical discourse. However, familiarity with certain terms related to integration and analysis is common. Below are related expressions in mathematical contexts:

1. "Lebesgue integrable" – A function is Lebesgue integrable if it meets the criteria for integration defined by the Lebesgue measure.

2. Translation: Функция является интегрируемой Лебега, если она соответствует критериям интеграции, определенным мерой Лебега.

3. "Absolutely integrable" – Refers to a function whose absolute value is integrable; a stronger condition than just being integrable.

4. Translation: Безусловно интегрируемая функция – это функция, абсолютное значение которой интегрируемо; это более сильное условие, чем просто быть интегрируемым.

Etymology

The term "integrable" derives from the root word "integrate," which comes from the Latin "integrare," meaning "to make whole." The prefix "everywhere" emphasizes the universality of this property across the entire domain of the function.

Synonyms and Antonyms

Synonyms: - Integrable - Finite integral

Antonyms: - Non-integrable - Divergent function

In summary, "everywhere integrable" is a technical term primarily used in mathematical literature to denote a function that can be integrated over its entire range, emphasizing its importance in theoretical and applied mathematics.