1 answer

4. Prove that the external measure is not "additive", in the sense that [0, 1]le <...

Question:

4. Prove that the external measure is not additive, in the sense that [0, 1]le < Vle+ |[0, 1] \Vle.
5. Let 7(x) := lim sup fx(x). +00 Prove that, for every a € R, {7 > a} = UN U{n>a + m}: MENJEN
4. Prove that the external measure is not "additive", in the sense that [0, 1]le < Vle+ |[0, 1] \Vle.
5. Let 7(x) := lim sup fx(x). +00 Prove that, for every a € R, {7 > a} = UN U{n>a + m}: MENJEN

Answers


4) It is sufficient to prove the result for V C[0, 1]. Define external measure Sle = inf l (Ik) where the infimum is taken ov

Now, consider Vº = [0, 1] \V = {0}. Also, consider a countable collection of open intervals C = { 1+ - where k EN Then clearl

.

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