Define μ ∗ ( E ) as the number of points in E if E is finite and μ ∗ ( E ) = ∞ if E is infinite. Show that μ ∗ is an outer measure. Determine the measurable sets.Evidently, μ ∗ is nonegative and monotone, and μ ∗ ( ∅ ) = 0. Let E n be a sequence of sets. It remains to show that μ ∗ is countably subadditive, i.e., (1) μ ∗ ( ⋃ n = 1 ∞ E n ) ≤ ∑ n = 1 ∞ μ ∗ ( E n ) . If ∑ μ ∗ ( E n ) = ∞, then there is nothing to prove. Suppose ∑ μ ∗ ( E n ) &lt; ∞. Then μ ∗ ( E n ) → 0, so that E n = ∅ for all values of n form some definite index N onward. Hence (1) reduces to (2) μ ∗ ( ⋃ n = 1 N − 1 E n ) ≤ ∑ n = 1 N − 1 μ ∗ ( E n ) . But (2) follows from the inclusion-exclusion principle. Therefore μ ∗ is an outer measure.That finite sets are measurable is obvious. What about infinite sets?

Question

Define       μ    ∗    (  E  ) as the number of points in   E if   E is finite and       μ    ∗    (  E  )  =  ∞ if   E is infinite. Show that       μ    ∗   is an outer measure. Determine the measurable sets.Evidently,       μ    ∗   is nonegative and monotone, and       μ    ∗    (  ∅  )  =  0. Let       E    n   be a sequence of sets. It remains to show that       μ    ∗   is countably subadditive, i.e.,                    (1)                              μ          ∗                (                  ⋃                      n            =            1                    ∞                          E          n                )        ≤                  ∑                      n            =            1                    ∞                          μ          ∗                (                  E          n                )                  .                    If   ∑      μ    ∗    (      E    n    )  =  ∞, then there is nothing to prove. Suppose   ∑      μ    ∗    (      E    n    )  &amp;lt;  ∞. Then       μ    ∗    (      E    n    )  →  0, so that       E    n    =  ∅ for all values of   n form some definite index   N onward. Hence (1) reduces to                    (2)                              μ          ∗                (                  ⋃                      n            =            1                                N            −            1                                    E          n                )        ≤                  ∑                      n            =            1                                N            −            1                                    μ          ∗                (                  E          n                )                  .                    But (2) follows from the inclusion-exclusion principle. Therefore       μ    ∗   is an outer measure.That finite sets are measurable is obvious. What about infinite sets?

Vitulloh0 · Accepted Answer

Let   X denote the set on which       μ    ∗   is defined. Then       μ    ∗   is additive on       P    (  X  ), and thus, every subset of   X is measurable.

Define μ ∗ </msup> ( E ) as the numbe

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