## 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 over all countable collections of open intervals {Ik} with the property that SCUIk and 1(I) denotes the length of the interval I. Now, consider V = {0} .

Also, consider a countable collection of open intervals Ck = { where k EN Then clearly, V CIk for each k EN. Now, l(Ik) = Then, \Vle = int {2}=0.

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

.
Now, l() = k/ Then, IV c = iờ {1+1} = 1. Similarly we can show that [0, 1]le = 1. Which shows that the external measure is not additive.

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