Interactive Real Analysis - part of

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Examples 7.3.7(a): Measurable Sets

Show that the empty set, the set R, and the complement of a measurable set are all measurable.
We have shown that the outer measure of the empty set 0 is zero, and sets with outer measure zero are automatically measurable.

For the set R of all real numbers we have:

m*(A R) + m*(A comp(R)) = m*(A) + m*(A 0) = m*(A)
which shows that R is measurable.

If a set E is measurable we have:

m*(A) = m*(A E) + m*(A comp(E))
For comp(E) we then have:
m*(A comp(E)) + m*(A comp(comp(E))) =
      = m*(A comp(E)) + m*(A E) =
      = m*(A)
which shows that comp(E) is measurable.
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