Assignment 5 : Assigned

  • Published 2012

Abstract

1. For each of the following sets, compute the Lebesgue outer measure. (a) Any countable set. (b) The Cantor set. (c) {x ∈ [0, 1] | x 6∈ Q}. 2. (a) If V ⊆ R is a subspace with dim(V ) < d, then show that λ(V ) = 0. (b) If P ⊆ R is a polygon show that area(P ) = λ(P ). 3. (a) Say μ is a translation invariant measure on (R,L) (i.e. μ(x+ A) = μ(A) for all A ∈ L, x ∈ R) which is finite on bounded sets. Show that ∃c > 0 such that μ(A) = cλ(A). (b) Let T : R → R be an orthogonal linear transformation, and A ∈ L. Show that T (A) ∈ L and λ(T (A)) = λ(A). [Hint: Express T in terms of elementary

Cite this paper

@inproceedings{2012Assignment5, title={Assignment 5 : Assigned}, author={}, year={2012} }