Re: generalized Lusin type thereom

craig wrote:

a friend is looking for a generalized Lusin type
theroem.

The regular one:

Suppose (X,d) a metric space and \mu a Borel
measure on
X which satisfies a few properties.

Then given f:X->R measurable there is some
continuous g
that is equal to f except on a set of small \mu
measure.

Generalized version:

He wants a version where f:X->X or maybe f:X-> Y
where
Y is another metric space.

Anyone heard of a version like this?

It's been a while since I've looked at Lusin's
theorem,
but I suspect if such a version exists it can be
found
in Stone's paper:

Arthur H. Stone, "Lusin's theorem", Atti del
Seminario Matematico
e Fisico dell'Universita di Modena 44 (1996),
351-357.

Unfortunately, I don't have a copy with me now (it's
at
home, and I'm not), so I can't tell you for sure. My
recollection is that Stone gave some fairly general
versions of Lusin's theorem, but the generalizations
were for the domain (as a certain topological space)
and for the "measures" defined on the domain.

Other places to look are listed in the post below.
The
second post below is a short essay on Lusin's
theorem,
but the focus there is on functions from R to R.

http://groups.google.com/group/sci.math/msg/f04e2f4466
d338f1

http://groups.google.com/group/sci.math/msg/680691c6ee
b50b91

Dave L. Renfro



thanks

craig
.