For a functor $F\supset Id$ on the category of metrizable compacta, we introduce a conception of a linear functorial operator $T=\{T_X:Pc(X)\to Pc(FX)\}$ extending (for each $X$) pseudometrics from $X$ onto $FX\supset X$ (briefly LFOEP for $F$). The main result states that the functor $SP^n_G$ of $G$-symmetric power admits a LFOEP if and only if the action of $G$ on $\{1,\dots,n\}$ has a one-point orbit. Since both the hyperspace functor $\exp$ and the probability measure functor $P$ contain $SP^2$ as a subfunctor, this implies that both $\exp$ and $P$ do not admit LFOEP.
@article{118932,
author = {Taras O. Banakh and Oleg Pikhurko},
title = {On linear functorial operators extending pseudometrics},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {38},
year = {1997},
pages = {343-348},
zbl = {0886.54010},
mrnumber = {1455501},
language = {en},
url = {http://dml.mathdoc.fr/item/118932}
}
Banakh, Taras O.; Pikhurko, Oleg. On linear functorial operators extending pseudometrics. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 343-348. http://gdmltest.u-ga.fr/item/118932/
AE(0)-spaces and regular operators extending (averaging) pseudometrics, Bull. Polon. Acad. Sci. Ser. Sci. Math. (1994), 42 197-206. (1994) | MR 1811849 | Zbl 0827.54010
On the spaces of measurable functions, Studia Math. 44 (1972), 597-615. (1972) | MR 0368068
On some geometric properties of covariant functors (in Russian), Uspekhi Mat. Nauk 39 (1984), 169-208. (1984) | MR 0764014
Triples of infinite iterates of metrizable functors (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. (1990), 54 396-418. (1990)
General Topology. Principal Constructions (in Russian), Moscow Univ. Press Moscow (1988). (1988)
Extending metrics in compact pairs, Mat. Studiï 3 (1994), 103-106. (1994) | MR 1692801 | Zbl 0927.54029
Regular linear operators extending metrics: a short proof, Bull. Polish. Acad. Sci. 44 (1996), 267-269. (1996) | MR 1419399 | Zbl 0866.54017