Metrically invariant measures on locally homogeneous spaces and hyperspaces

Functionals on the spaces of convex bodies

“…Just like the Euclidean case, we call {K : In 1986 Bandt and Baraki [1] proved the following result, which in certain sense gave a negative answer to this problem. When n ≥ 2, there is no positive δ-finite Borel measure on {K n , · H } which is invariant with respect to all isometries in it.…”

Normally Distributed Probability Measure on the Metric Space of Norms

“…Unfortunatelly this is not so. Bandt and Baraki in [1] proved answering to a problem of McMullen [15] that there is no positive σ-finite Borel measure on it which is invariant with respect to all isometries of (K, δ h ) into itself. This result exclude the possibility of the existence of a natural volume-type measure.…”

Cantor set approximations and dimension computations in hyperspaces.

“…Curtis and Schori developing their theory, Boardman was developing the following early measure theoretical results concerning hyperspaces (see [7], [8] Despite this early progress, hyperspaces remained untouched in a measuretheoretical sense for over a decade, until Bandt and Baraki developed the following powerful theorem in [2].…”

“…2 The simplest situation arises when the IFS maps are similitudes whose images satisfy a disjointness condition known as the strong separation condition (SSC). We formally define these two concepts with the following definitions.…”

“…The second motivating factor has to do with Bandt's theorem in [2]. In recent work, Elekes and Keleti have begun to study so-called dirnensionless or immeasurable compact sets in JR (see [15], [20]).…”