Interactive Real Analysis
- part of 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?
Context
This function is clearly defined for all subsets, and it is non-negative. As for
subadditive, it is certainly true that, for example:
Context2 = f({a}or{b, c})
f({a}) + f({b,c}) = 1 + 1 = 2
1 = f({a}but to prove general (countable) subadditivity requires just a little more thought. Do you feel up to it?