## Example 7.3.5(d): Properties of Outer Measure

Define a function

This function is clearly defined for all subsets, and it is non-negative. As for
subadditive, it is certainly true that, for example:
*f*on the set*by setting**= {a, b, c}***X***f(*,*) = 0***O***f(*, and*) = 2***X***f(*for any other subset*) = 1***A***of***A***. Could this function be called an outer measure, i.e. is it a non-negative set function whose domain is***X***and which is subadditive? Is it additive?***P(R)**or2 = f({a} {b, c}) f({a}) + f({b,c}) = 1 + 1 = 2

but to prove general (countable) subadditivity requires just a little more thought. Do you feel up to it?1 = f({a} {c}) f({a}) + f({c}) = 1 + 1 = 2