Stat 310A/Math 230A Theory of Probability Midterm Solutions
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چکیده
with a1, a2 > 0. (Indeed any rank 2 matrix can be written as RA with A as above, and R orthogonal. Hence any affine transformation can be written as h ◦ f with f as claimed and g a rigid motion.) Hence, by Caratheodory uniqueness theorem it is sufficient to show that λ2(ASu(α1, α2)) = |det(A)|λ2(Su(α1, α2)) for all u = (u1, u2) ∈ R, αi ∈ R+ (3) Su(α, β) ≡ { x = (x1, x2) ∈ R : x1 ∈ (u1, u1 + α1], x2 ∈ (u2, u2 + α2] } . (4)
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Stat 310A/Math 230A Theory of Probability Practice Midterm Solutions
Solution : Throughout the solution we will use the fact that λ2 = λ1 × λ1 whence we obtain the action of λ2 on rectangles: λ2(A1 × A2) = λ1(A1)λ1(A2). Also, for J1, J2 ⊆ R two intervals, let TJ1,J2 be any triangle with two sides equal to J1 (parallel to the first axis) and J2 (equal to the second axis). from the addittivity of λ2 it follows immediately that λ2(TJ1,J2) = |J1| · |J2|/2. (We use h...
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