Let A ⊂ R be Lebesgue measurable such that 0 &lt; λ ( A ) &lt; ∞ and consider the function f : R → [ 0 , ∞ ), f ( x ) = λ ( A ∩ ( − ∞ , x ) ). I would like to compute lim x → ∞ f ( x ).I have already proved that f is continuous (I have in fact showed that it is Lipschitz continuous), but I am not really sure if this allows me to say that lim x → ∞ f ( x ) = λ ( A ∩ ( − ∞ , ∞ ) ) = λ ( A ). Intuitively, I am sure that this is right, but I am not sure whether the fact that f is continuous is enough to write this. I feel that I am somehow assuming that the Lebesgue measure λ is continuous in some sense. Could you please tell me if my conclusion is right and if the mere continuity of f is enough for it?

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Let   A  ⊂      R   be Lebesgue measurable such that   0  &amp;lt;  λ  (  A  )  &amp;lt;  ∞ and consider the function   f  :      R    →  [  0  ,  ∞  ),   f  (  x  )  =  λ  (  A  ∩  (  −  ∞  ,  x  )  ). I would like to compute       lim          x      →      ∞        f  (  x  ).I have already proved that f is continuous (I have in fact showed that it is Lipschitz continuous), but I am not really sure if this allows me to say that       lim          x      →      ∞        f  (  x  )  =  λ  (  A  ∩  (  −  ∞  ,  ∞  )  )  =  λ  (  A  ). Intuitively, I am sure that this is right, but I am not sure whether the fact that   f is continuous is enough to write this. I feel that I am somehow assuming that the Lebesgue measure   λ is continuous in some sense. Could you please tell me if my conclusion is right and if the mere continuity of   f is enough for it?

Govorei9b · Accepted Answer

Of course you can write this. The reason is the following: if   x  &amp;lt;      x    ′  , then   f  (  x  )  ≤  f  (      x    ′    ) so   f is increasing. Also,   f is bounded above by   λ  (  A  ), so       lim          x      →      ∞        f  (  x  ) exists. Therefore, the limit is equal to       lim          n      →      ∞        λ  (  A  ∩  (  −  ∞  ,  n  )  ). Now since   A  ∩  (  −  ∞  ,  n  )  ⊂  A  ∩  (  ∞  ,  n  +  1  ) for all   n, we have   λ  (  A  )  =  λ  (      ⋃          n      ≥      1        (  A  ∩  (  −  ∞  ,  n  )  )  )  =      lim          n      →      ∞        λ  (  A  ∩  (  −  ∞  ,  n  )  ), where we used the following well-known fact:If   μ is a measure and   (      A    n    ) is an increasing sequence of measurable sets, then   μ  (      ⋃    n        A    n    )  =      lim          n      →      ∞        μ  (      A    n    ).

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