The Riemann integrable functions are \(\mathbf\Pi_{3}^{0}\)-complete in the Lebesgue integrable functions (Q882047)
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scientific article; zbMATH DE number 5156455
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The Riemann integrable functions are \(\mathbf\Pi_{3}^{0}\)-complete in the Lebesgue integrable functions |
scientific article; zbMATH DE number 5156455 |
Statements
The Riemann integrable functions are \(\mathbf\Pi_{3}^{0}\)-complete in the Lebesgue integrable functions (English)
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23 May 2007
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The goal of the paper is to study the class of bounded Riemann integrable functions. Instead of this class the author considers the wider class \(\mathcal R\) of all measurable functions on \([0,1]\) which are equal almost everywhere to some bounded Riemann integrable function. The main result of the paper states that \(\mathcal R\) is a~meager \(\Pi^0_3\)-complete subset of \(L^p[0,1]\) for each \(1\leq p<\infty\).
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Borel sets
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Riemann integrable functions
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\(\Pi^0_3\)-complete
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\(L^p\)-space
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Borel complexity
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