Some properties of m-density points and density-degree functions are studied. Moreover, the following main results are provided:Let lambda be a continuous differential form of degree h in R-n (with h &gt;= 0) having the following property: there exists a continuous differential form Delta of degree h + 1 in R-n such thatintegral(Rn Delta boolean AND omega = integral Rn lambda boolean AND d omega,)for every C-c(infinity) differential form omega of degree n - h - 1 in R-n. Moreover let mu be a C-1 differential form of degree h + 1 in R-n and set E := {y is an element of R-n vertical bar Delta(y) = mu(y)}. Then d mu(x) = 0 whenever x is a (n + 1)-density point of E.Let f : R-n -&gt; (R) over bar be a measurable function such that f(x) is an element of {0} boolean OR [n, +infinity] for a.e. x is an element of R-n. Then there exists a countable family {F-k}(k=1)(infinity) of closed subsets of R-n such that the corresponding sequence of density-degree functions {d(Fk)}(k=1)(infinity) converges almost everywhere to f.

A closure result for differential forms in the context of superdensity, approximation of measurable functions by density-degree functions / Delladio, S. - In: BANACH JOURNAL OF MATHEMATICAL ANALYSIS. - ISSN 2662-2033. - 16:3(2022), pp. 4001-4019. [10.1007/s43037-022-00192-8]

### A closure result for differential forms in the context of superdensity, approximation of measurable functions by density-degree functions

#### Abstract

Some properties of m-density points and density-degree functions are studied. Moreover, the following main results are provided:Let lambda be a continuous differential form of degree h in R-n (with h >= 0) having the following property: there exists a continuous differential form Delta of degree h + 1 in R-n such thatintegral(Rn Delta boolean AND omega = integral Rn lambda boolean AND d omega,)for every C-c(infinity) differential form omega of degree n - h - 1 in R-n. Moreover let mu be a C-1 differential form of degree h + 1 in R-n and set E := {y is an element of R-n vertical bar Delta(y) = mu(y)}. Then d mu(x) = 0 whenever x is a (n + 1)-density point of E.Let f : R-n -> (R) over bar be a measurable function such that f(x) is an element of {0} boolean OR [n, +infinity] for a.e. x is an element of R-n. Then there exists a countable family {F-k}(k=1)(infinity) of closed subsets of R-n such that the corresponding sequence of density-degree functions {d(Fk)}(k=1)(infinity) converges almost everywhere to f.
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2022
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Delladio, S
A closure result for differential forms in the context of superdensity, approximation of measurable functions by density-degree functions / Delladio, S. - In: BANACH JOURNAL OF MATHEMATICAL ANALYSIS. - ISSN 2662-2033. - 16:3(2022), pp. 4001-4019. [10.1007/s43037-022-00192-8]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11572/391092`
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