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Examples 7.3.7(c): Measurable Sets

Show that the union of two measurable sets is measurable.
Assume that E and F are two measurable sets. Then

We need to prove: for every set A we have

m*(A) m*(A (E F) + m*(A comp(E F))
We know:
  1. E is measurable so that
    m*(A) = m*(A E) + m*(A comp(E))
    for every set A
  2. F is measurable so that
    m*(A) = m*(A F) + m*(A comp(F))
  3. From set theory:
    A (E F) = (A E) (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))
From (3) we have:
m*(A (E F)) m*(A E) + m*(A comp(E) F)
Adding m*(A comp(E F)) on both sides gives:
m*(A (E F)) + m*(A comp(E F))
      m*(A E) + m*(A comp(E) F) + m*(A comp(E F))
The last two terms add up to m*(A comp(E)) so that
m*(A (E F)) + m*(A comp(E F))
      m*(A E) + m*(A comp(E)) = m*(A)
because of (1). But then we have
m*(A) m*(A (E F)) + m*(A comp(E F))
     
which is exactly what we had to show - proof finished (not very enlightning, but done).
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