mention that we use in the first then using m for measurement.
the length function extends this concept to a function
we need this measure
- stay unchanged under translation
-
this is the goal we want to define a measurement, but today we will show it's impossible.
take Vitali set into consideration:
if
and
.
stands for the equivalence class of x.take
.Take
into consideration,which means every
,there is one and only one element
satisfies
~
.So for all
,so
so
,a hilarious thing!
So it's impossible to define a function with above 4 properties. Let's try to see how far we can go.To see the biggest part of the subset of real numbers which can define a good measure function.
