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Counterexample of limsup of sets and measures

Mathematics Asked by David Warren Katz on August 7, 2020

On page 163 in Wheeden-Zygmund, it is proved that for a nonnegative additive set function $phi$, $ overline{lim} phi(A_n) le phi(overline{lim} A_n)$ for any sequence of measurable functions $(A_n)$, where $overline{lim}$ denotes the lim sup. I know that, intuitively, volume can disappear at infinity (example below), so $phi(underline{lim} A_n) le underline{lim} phi(A_n)$ is intuitive to me. However, $ overline{lim} phi(A_n) le phi(overline{lim} A_n)$ does not make intuitive sense to me. I cannot find a problem with the proof in the text, but I also have found what may be a counterexample.

Let $A_n = [0, n] times [0, frac{1}{n}] subseteq mathbb{R}^2$, and let
begin{equation}
phi(A)=
begin{cases}
m(A) &text{ if} quad m(A) < infty \
0, & text{otherwise}
end{cases}
end{equation}

where $m$ denotes the Lebesgue measure of $mathbb{R}^2$. It follows that $phi$ is a nonegative additive set function on measure space ${(mathbb{R}^2, M)}$, where $M$ is the Lebesgue measurable sets.

Now, note that $overline{lim} phi(A_n) = 1$, since $phi(A_n) = 1$ for every $n$. However, $overline{lim}A_n = mathbb{R}$, so $phi(overline{lim}A_n) = 0$, contradicting $ overline{lim} phi(A_n) le phi(overline{lim} A_n)$.

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