MathCS.org - Real Analysis: Examples 7.3.7(a): Measurable Sets

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.

Context

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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