I am taking a course in Integration theory right now. A random fun fact I encountered in the notes to the book we are using, “Real Analysis” by Folland, is that you can actually extend the Lebesgue measure to a larger sigma-algebra than the usual Lebesgue sigma-algebra in such a way that the extended measure remains translation invariant.
If we consider the Hilbert-space of square-integrable functions with this new measure it is quite a bit larger than the normal space 
. The new space will have dimension equal to the cardinality of the real numbers. (That is, any basis of the space has cardinality equal to the real numbers.)
One can extend Lebesgue measure even further if one only insists on the new “measure” being finitely additive. One can define a function, 
, from the entire power set 
to ![[0,\infty]](http://s0.wp.com/latex.php?latex=%5B0%2C%5Cinfty%5D&bg=ffffff&fg=333333&s=0 "[0,\infty]")
such that it is translation-invariant, finitely additive (that is 
if 
and 
are disjoint) and coincides with the Lebesgue measure on all measurable sets.
On 
one can define a finitely additive measure that is translation-invariant and rotation-invariant and is an extension of Lebesgue measure. But on 
the Banach-Tarski paradox shows that no such thing can be done.
Like this:
Be the first to like this post.
Tags: Integration theory, Lebesgue measure, Real analysis
This entry was posted on 2010/03/09 at 6:38 pm and is filed under Math. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.
