Math 2210 Real Analysis 1 Problem Set 4
نویسنده
چکیده
Solution. Let X = E ∪ F with E,F ∈ M such that E ∩ F = ∅, E is null for ν−, and F is null for ν. Let U ⊂ X. If U ⊂ E then ν(U) = ν(U) = λ(U) − μ(U) ≤ λ(U), so λ(U) ≥ ν(U). Similarly, if U ⊂ F then μ(U) ≥ ν−(U). Now if V = U ∩ E and W = U ∩ F then V and W are measurable and ν(U) = ν(V ) ≤ λ(V ) (by the previous argument) ≤ λ(U) since λ is a positive measure. Similarly, ν−(U) = ν−(W ) ≤ μ(W ) ≤ μ(U).
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