The topology of close approximation
Let \({\mathcal L}\subset R^ R\) denote the set of all Lebesgue measurable real-valued functions defined on R, and let \({\mathcal L}^+\) denote \(\{f\in {\mathcal L}:\) \(f(x)>0\) for all \(x\in R\}\). The topology of close approximation, \({\mathcal T}_ c\), is defined on \(R^ R\) by taking as a neighbourhood base of \(f\in R^ R\) the sets \(\hat C(f,\epsilon)=\{g\in R^ R|\) \(g(x)-f(x)| <\epsilon (x)\) for all \(x\in R\}\), where \(\epsilon\in {\mathcal L}^+\). The uniform topology, \({\mathcal T}_ u\), is defined on \(R^ R\) by taking as neighbourhood base of \(f\in R^ R\) the sets \(\hat U(f,\epsilon)=(g\in R^ R:| g(x)-f(x)| <\epsilon\) for all \(x\in R\}\), where \(\epsilon \in R^+\), the set of positive real numbers. The Lusin topology, \({\mathcal T}_ L\), of convergence in measure is defined by the neighborhood base of \(f\in R^ R\) \(\hat L(f,\epsilon)=\{g\in R^ R:\lambda^*\{x\in R| g(x)-f(x)| \geq \epsilon \}<\epsilon\}\), where \(\epsilon \in R^+\) and \(\lambda^*\) denotes the Lebesgue outer measure. Clearly \({\mathcal T}_ L\subset {\mathcal T}_ u\subset {\mathcal T}_ c.\) The authors discuss the properties of the spaces (\({\mathcal L},{\mathcal T}_ c)\), (\({\mathcal L},{\mathcal T}_ u)\) and (\({\mathcal L},{\mathcal T}_ L)\) interacting on each other, and the situation of their subspaces, which are summarized as below: 1. \({\mathcal L}\) is a closed \(G_{\delta}\)-set in \(R^ R\), in each of \({\mathcal T}_ L\), \({\mathcal T}_ u\) and \({\mathcal T}_ c.\) Let \({\mathcal L}'\) be the set \(\{\) \(f\in {\mathcal L}:\int_{R}| f(x)| d\lambda(x)<\infty\}\) and \({\mathcal C}\) the set of all continuous functions from R to R. 2. All four spaces \(R^ R\), \({\mathcal L}\), \({\mathcal L}'\) and \({\mathcal C}\) are Baire spaces in all three topologies \({\mathcal T}_ L\), \({\mathcal T}_ u\) and \({\mathcal T}_ c\), except for (\({\mathcal L}',{\mathcal T}_ L)\), (\({\mathcal L}',{\mathcal T}_ u)\) and (\({\mathcal C},{\mathcal T}_ L).\) 3. (\({\mathcal C},{\mathcal T}_ c)\) is discrete. 4. (Miller) The character of (\({\mathcal L},{\mathcal T}_ c)\) is \(2^ c.\) 5. The following functions form a closed nowhere dense set in (\({\mathcal L},{\mathcal T}_ c)\). (a) surjective measurable functions, (b) measurable function that are continuous at least one point, (c) measurable functions that are somewhere of BV (bounded variation on some nonempty interval). 6. The one-to-one continuous functions form a nowhere dense \(G_{\delta}\)-set in (\({\mathcal C},{\mathcal T}_ u)\) and the one-to-one functions are dense in (\({\mathcal L},{\mathcal T}_ c)\), the following are equivalent: (i) K is countably compact, (ii) K is compact, (iii) K is compact and metrizable and (iv) K is finite union \(K_ 1\cup K_ 2\cup...\cup K_ n\), where each \(K_ i\) is a compact subset of \(L_ s(h_ i)\) (in its ordinary Euclidean topology) for some finite \(S_ i\subset R\) and some \(h_ i\in {\mathcal L}.\) For \(f,g\in R^ R\), let \(f\sim g\) be \(\lambda\{x\in R:\) \(f(x)\neq g(x)\}=0\) and [f] the equivalence class of f under this equivalence relation, and let \({\mathcal E}\) denote the set of equivalence classes \(\{[f]: f\in {\mathcal L}\}\). The topologies \({\mathcal T}'_ L\), \({\mathcal T}'_ u\) and \({\mathcal T}'_ c\) on \({\mathcal E}\) are defined in natural sense corresponding to \({\mathcal T}_ L\), \({\mathcal T}_ u\) and \({\mathcal T}_ c\), which are denoted by \({\mathcal T}'_ L\), \({\mathcal T}'_ u\) and \({\mathcal T}'_ c\). Clearly \({\mathcal T}'_ L\subset {\mathcal T}'_ u\subset {\mathcal T}'_ c.\) 8. \({\mathcal E}\) is a Baire space in each of \({\mathcal T}'_ L\), \({\mathcal T}'_ u\) and \({\mathcal T}'_ c.\) 9. (MA) the character of (\({\mathcal E},{\mathcal T}'_ c)\) is c. 10. (CH) (\({\mathcal E},{\mathcal T}'_ c)\) is hereditarily paracompact.
Lebesgue measurable real-valued functions
scientific article; zbMATH DE number 4178081
scientific article