Some examples of measurable norms

Some examples and the connection between cylindrical measures and measurable norms

“…But the converse is false in general ( [5], this is the 1984-Example). If µ is a generalized quasi-rotationally invariant cylindrical measure, then the converse is true [6].…”

“…Now choose a number δ > 0 in (5) such that η 1 + 2η 2 + · · · + (n + m)η n+m < 1 and 0 < η 2 1 + · · · + η 2 n+m < 1. Then we have 4N < 2 η 1 + 2η 2 + · · · + (n + m)η n+m + (2N + 1) < 4(N + 1), and N < N η 2 1 + · · · + η 2 n+m + 1 < 2N.…”

“…The first author solved this problem negatively [5]. Recently we find that the norm in the 1984-Example is not γ -measurable, where γ is the canonical Gauss cylindrical measure.…”

Measurable norms and rotationally quasi-invariant cylindrical measures

“…However, the converse was the open problem ( [1]). Recently the author solved it negatively ( [10]).…”