Page 264 - DMTH503_TOPOLOGY

Topology

Notes 3. If dim (X) < d then show that the (d-dimensional) Lebesgue measure of X is zero.
H
4. Let  ,   and let
1 2
 +  = { +  :   ,   }
1 2 1 2 1 1 2 2
then prove that dim ( +  )dim ( ) + dim ( ).
H 1 2 H 1 H 2
5. If we can find a probability measure  satisfying the above hypothesis then prove that
dim (X)³d.
H
32.5 Further Readings

Books Rogers, M. (1998), Hausdorff Measures, Cambridge University Press.

Lapidus, M. (1999), Math 209A – Real Analysis Mid-term, UCR Reprographics.

Online links en.wikipedia.org/wiki/E8-mathematics
en.wikipedia.org/wiki/M-theory

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