1.
If u is an outer measure and if every open set is measuable,
then u is a metric outer measure.
2.
Consider the transformation Tx=Ax+k in R^n, where A is a nonsingular
n*n matrix and x, k are column n-vectors. T maps sets E onto sets T(E).
Assume that 入[T(I(a,b))] = |detA|入(I(a,b)). Prove that satisfies the
properties (a)-(c).
(a) For any set E, u(T(E))=|detA|u(E), u is outer measure.
(b) E is lebesgue-measurable if and only if T(E) is lebesgue-measurable.
(c) If E is lebesgue-measurable, then u(T(E))=|detA|u(E), u is lebesgue
measure.
3.
Given an example of a signd measure for which the Hahn decomposition is not
unique.
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