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Problem 4. Let Λ be a diagonal matrix, with all λi,i > 0. Consider a measurable...

Problem 4. Let Λ be a diagonal matrix, with all λi,i > 0. Consider a measurable E ⊂ Rd .

Define ΛE = {Λx : x ∈ E}.

Prove that ΛE is measurable and m(ΛE) = [det(Λ)]m(E).

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Answer #1

is a diagonal martix with diagonal entries . Hence the matrix is invertible.

is a continuous function, hence measurable. Let be measurable, then is measurable.

is measurable.

Let be a measurable rectangle in .

.

But

Since is true for all measurable rectangles and is a constant.

for any measurable set.

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