Examples 7.3.7(c): Measurable Sets
Assume that E and F are two measurable sets. We need to prove that for every set A we have:
m*(A) m*(A (E F) + m*(A comp(E F))
We know that
- E is measurable so that for every set A we have:
m*(A) = m*(A E) + m*(A comp(E))
- F is measurable so that for every set A we have:
m*(A) = m*(A F) + m*(A comp(F))
- From set theory we know (draw a Venn diagram to verify) that:
A (E F) = (A E) (A comp(E) F)
which implies by subadditivity of m* that
m*(A (E F)) m*(A E)) + m*(A comp(E) F))
Using A comp(E) in place of A in (2) gives:
m*(A comp(E)) =
= m*(A comp(E) F) + m*(A comp(E) comp(F)) =
= m*(A comp(E) F) + m*(A comp(E F))
We can now substitute that into (1) to get:
m*(A) = m*(A E) + m*(A comp(E) F) + m*(A comp(E F))
m*(A (E F)) + m*(A comp(E F))
Of course we used (3) to obtain the inequality. But that's what we wanted to show: proof finished (not very enlightning, but done).
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