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**The measures of nonconvex total boundedness and of nonstrongly convex total boundedness for subsets of \(L_0\).**
*(English)*
Zbl 1047.46004

The extension of the Schauder-Tikhonov fixed point property to nonlocally convex linear spaces is an open problem, even in \(L_0\), the space of measurable functions. [Editorial comment: This has meanwhile be proved in the affirmative, in full generality, by R. Cauty in Fundam. Math. 170, No. 3, 231–246 (2001; Zbl 0983.54045).] A known sufficient condition for a compact convex subset to have the fixed point property is that the set have the property of convex total boundedness. The author, E. de Pascale and H. Weber [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 20, No. 3, 341–355 (1993; Zbl 0805.47055)] introduced the concept of strongly convex total boundedness (SCTB) and introduced parameters that measure the degree of nonstrongly convex total boundedness. The concept is invariant when one passes to the convex hull. The notions of strongly convex total boundedness and the noncompactness measures were used to prove a Fan’s best approximation theorem in nonlocally convex linear spaces. In [H. Weber, Note Mat. 12, 271–289 (1992; Zbl 0846.46004)] SCTB was linked to affine embeddability in locally convex linear spaces.

Here the author gives estimates of the measures of nonconvex total boundedness and of nonstrongly convex total boundedness in \(L_0\), the closure of the space of simple functions \(S\), with underlying space \(\Omega\), measure \(\eta\), and pseudonorm \(\| f\| _0 = \inf \{ \alpha > 0 \mid \eta(\{| f| \geq \alpha \} ) \leq \alpha \}\). For example, Fréchet and Smulian proved that a subset \(M\) of \(T_0(\Omega)\), the space of real valued totally measurable functions defined on \(\Omega \subseteq \mathbb{R}^n\) is relatively compact with respect to the topology of convergence in measure if and only if the set is equally measurable and equally quasi-bounded (see below). The parameters that connect this are \[ \gamma(M) = \inf \{ \varepsilon > 0 \mid \text{ there is a finite subset } F \text{ such that } M \subseteq F + B_{\varepsilon} \}, \] where \(B_{\varepsilon}= \{ \| f\| _0 \leq \varepsilon \}\) is the \(\varepsilon\)-pseudoball, \[ \lambda_0(M) = \inf \{ \varepsilon > 0 \mid \text{ there is an } a \in [0, +\infty) \text{ such that } M \subseteq [-a,a] + B_{\varepsilon} \}, \] where \([-a,a]\) are the functions in \(L_0\) with \(| f| \leq a\), and \[ \begin{split} k_0(M) = \inf \{ \varepsilon > 0 \mid \text{ there is a partition } A_1, \ldots, A_n \text{ of } \Omega\\ \text{ such that } M \subseteq S(A_1, \ldots, A_n) + B_{\varepsilon} \},\end{split} \] where \(S(A_1, \ldots, A_n) = \{ s \mid s = \sum_1^n \alpha_i \chi_{A_i} \}\) are the simple functions.

With these parameters, the author shows that \(M \subseteq T_0(\Omega)\) is equally quasi-bounded if and only if \(\lambda_0(M) = 0\), and is equally measurable if and only if \(k_0(M) = 0\). Since clearly \(M\) is compact if and only if \(\gamma(M) =0\), the Frechet-Smulian theorem then follows from Theorem 2.1: \(\max(\lambda_0(M), k_0(M)) \leq \gamma(M) \leq \lambda_0(M) + k_0(M)\). He introduces similar parameters involving convex sets to characterize strongly convex totally bounded sets and to characterize when a set is affinely embeddable in a locally convex space. For example, \[ \begin{split} \lambda_s(M) = \inf \{ \varepsilon > 0 \mid \text{ there is an } a \in [0, +\infty) \text{ and a convex subset }\\ C \text{ of } B_{\varepsilon} \text{ such that } M \subseteq [-a,a] + C \} \end{split} \] and \[ \begin{split} \omega_s(M) = \inf \{ \varepsilon > 0 \mid \text{ there is a partition } A_1, \ldots, A_n \text{ of } \Omega \text{ and a convex subset }\\ C \text{ of } B_{\varepsilon} \text{ such that } M \subseteq S(A_1, \ldots, A_n) + C \},\end{split} \] and he gives estimates for these and other parameters that imply, for example, that a convex subset \(M\) of \(L_0\) is SCTB if and only if \(\lambda_0(M) = \omega_s(M) = 0\), and a convex compact subset of \(L_0\) is SCTB if and only if it is affinely embeddable in a locally convex space if and only if \(\lambda_s(M) = \omega_s(M) = 0\).

Unfortunately, a number of possibly confusing misprints (such as the appearance of \(L-0\) on page 193 (probably \(L_0\)), omission of convex in a key part of the definition of a locally convex subset, writing \(M \subseteq [-a,a] + B\) instead of \(M \subseteq [-a,a] + C\) with the convex set \(C\) in the definition of \(\lambda_s\), and the omission of the subscripts \(_s\) on page 202) mar this very interesting paper.

Here the author gives estimates of the measures of nonconvex total boundedness and of nonstrongly convex total boundedness in \(L_0\), the closure of the space of simple functions \(S\), with underlying space \(\Omega\), measure \(\eta\), and pseudonorm \(\| f\| _0 = \inf \{ \alpha > 0 \mid \eta(\{| f| \geq \alpha \} ) \leq \alpha \}\). For example, Fréchet and Smulian proved that a subset \(M\) of \(T_0(\Omega)\), the space of real valued totally measurable functions defined on \(\Omega \subseteq \mathbb{R}^n\) is relatively compact with respect to the topology of convergence in measure if and only if the set is equally measurable and equally quasi-bounded (see below). The parameters that connect this are \[ \gamma(M) = \inf \{ \varepsilon > 0 \mid \text{ there is a finite subset } F \text{ such that } M \subseteq F + B_{\varepsilon} \}, \] where \(B_{\varepsilon}= \{ \| f\| _0 \leq \varepsilon \}\) is the \(\varepsilon\)-pseudoball, \[ \lambda_0(M) = \inf \{ \varepsilon > 0 \mid \text{ there is an } a \in [0, +\infty) \text{ such that } M \subseteq [-a,a] + B_{\varepsilon} \}, \] where \([-a,a]\) are the functions in \(L_0\) with \(| f| \leq a\), and \[ \begin{split} k_0(M) = \inf \{ \varepsilon > 0 \mid \text{ there is a partition } A_1, \ldots, A_n \text{ of } \Omega\\ \text{ such that } M \subseteq S(A_1, \ldots, A_n) + B_{\varepsilon} \},\end{split} \] where \(S(A_1, \ldots, A_n) = \{ s \mid s = \sum_1^n \alpha_i \chi_{A_i} \}\) are the simple functions.

With these parameters, the author shows that \(M \subseteq T_0(\Omega)\) is equally quasi-bounded if and only if \(\lambda_0(M) = 0\), and is equally measurable if and only if \(k_0(M) = 0\). Since clearly \(M\) is compact if and only if \(\gamma(M) =0\), the Frechet-Smulian theorem then follows from Theorem 2.1: \(\max(\lambda_0(M), k_0(M)) \leq \gamma(M) \leq \lambda_0(M) + k_0(M)\). He introduces similar parameters involving convex sets to characterize strongly convex totally bounded sets and to characterize when a set is affinely embeddable in a locally convex space. For example, \[ \begin{split} \lambda_s(M) = \inf \{ \varepsilon > 0 \mid \text{ there is an } a \in [0, +\infty) \text{ and a convex subset }\\ C \text{ of } B_{\varepsilon} \text{ such that } M \subseteq [-a,a] + C \} \end{split} \] and \[ \begin{split} \omega_s(M) = \inf \{ \varepsilon > 0 \mid \text{ there is a partition } A_1, \ldots, A_n \text{ of } \Omega \text{ and a convex subset }\\ C \text{ of } B_{\varepsilon} \text{ such that } M \subseteq S(A_1, \ldots, A_n) + C \},\end{split} \] and he gives estimates for these and other parameters that imply, for example, that a convex subset \(M\) of \(L_0\) is SCTB if and only if \(\lambda_0(M) = \omega_s(M) = 0\), and a convex compact subset of \(L_0\) is SCTB if and only if it is affinely embeddable in a locally convex space if and only if \(\lambda_s(M) = \omega_s(M) = 0\).

Unfortunately, a number of possibly confusing misprints (such as the appearance of \(L-0\) on page 193 (probably \(L_0\)), omission of convex in a key part of the definition of a locally convex subset, writing \(M \subseteq [-a,a] + B\) instead of \(M \subseteq [-a,a] + C\) with the convex set \(C\) in the definition of \(\lambda_s\), and the omission of the subscripts \(_s\) on page 202) mar this very interesting paper.

Reviewer: Raymond Johnson (College Park)

### MSC:

46A16 | Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) |

47H10 | Fixed-point theorems |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

46A55 | Convex sets in topological linear spaces; Choquet theory |