# Interactive Real Analysis

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## 7.3. Measures

### 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. 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

1. E is measurable so that for every set A we have:
m*(A) = m*(A E) + m*(A comp(E))
2. F is measurable so that for every set A we have:
m*(A) = m*(A F) + m*(A comp(F))
3. 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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