7.3. Measures

Example 7.3.5(d): Properties of Outer Measure

Define a function f on the set X = {a, b, c} by setting f(O) = 0, f(X) = 2, and f(A) = 1 for any other subset A of X. Could this function be called an outer measure, i.e. is it a non-negative set function whose domain is P(R) and which is subadditive? Is it additive?
This function is clearly defined for all subsets, and it is non-negative. As for subadditive, it is certainly true that, for example:
2 = f({a} {b, c}) f({a}) + f({b,c}) = 1 + 1 = 2
or
1 = f({a} {c}) f({a}) + f({c}) = 1 + 1 = 2
but to prove general (countable) subadditivity requires just a little more thought. Do you feel up to it?
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