I am a masters student and I have taken a course on measure theory ( it is my 1 st measure theory course).
In one of initial classes professor gave defination of Haursdorff measure similar to as given here -- https://en.m.wikipedia.org/wiki/Hausdorff_measure.
and then immediately he gave assignment problem to find the Haursdorff measure of Cantor Dust and
Sierpinski carpet given here --- https://en.m.wikipedia.org/wiki/Sierpinski_carpet.
I could not solve it and found while searching on internet that Haursdorff measure of Sierpinski carpet has only been found in 1998 by Chinese mathematicians in a 8 page proof devoted entirely to it.
My professor never discusses any any assignment problems. In fact, he doesn't even answer doubts of students in class. So, I can't ask him.
I am thinking that 8 page proof is long enough for assignment problems. In fact, none of other problems were so long.
So, can someone please tell a somewhat not so lenthy approach and which doesn't uses much advanced concepts in measure theory to find Haursdorff measure of both sierpinski carpet and Cantor dust as this problem was given in initial lectures of 1 st course on measure theory.