Let ( Ω , A ) be a measurable space, and let μ : A → [ 0 , + ∞ ) be a monotone subadditive set function. Let moroever A 1 , A 2 , … ∈ A be a sequence of measurable sets. It is easy to prove that μ ( A n ) → 0 ⇒ μ ( A n c ) → μ ( Ω ). On the contrary, I am not able neither to prove the opposite implication, namely μ ( A n ) → μ ( Ω ) ⇒ μ ( A n c ) → 0, nor to find an example violating it. Can someone find either one or the other?

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Let   (  Ω  ,      A    ) be a measurable space, and let   μ  :      A    →  [  0  ,  +  ∞  ) be a monotone subadditive set function. Let moroever       A    1    ,      A    2    ,  …  ∈      A   be a sequence of measurable sets. It is easy to prove that   μ  (      A    n    )  →  0  ⇒  μ  (      A    n    c    )  →  μ  (  Ω  ). On the contrary, I am not able neither to prove the opposite implication, namely   μ  (      A    n    )  →  μ  (  Ω  )  ⇒  μ  (      A    n    c    )  →  0, nor to find an example violating it. Can someone find either one or the other?

Sophia Mcdowell · Accepted Answer

It&#039;s not true that   μ  (      A    n    )  →  μ  (  Ω  )    ⟹    μ  (      A    n    c    )  →  0. Consider       R   endowed with the Lebesgue σ-algebra. Let       A    n    =  [  0  ,  n  ]. Then   μ  (      A    n    )  =  n  →  +  ∞  =  μ  (      R    ), but   μ  (      A    n    c    )  =  +  ∞ for all   n.

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