In measure theory, given sets $A_1,A_2,\ldots$, we define $\liminf A_n=\bigcup_{k=1}^\infty\left(\bigcap_{n\geq k}A_n\right)$ and $\limsup A_n=\bigcap_{k=1}^\infty\left(\bigcup_{n\geq k}A_n\right)$.
What is the relation to the normal liminf/limsup for sequences? $\liminf{a_n} = \lim_{n\rightarrow\infty}\inf(a_n,a_{n+1},\ldots)$ and $\limsup{a_n} = \lim_{n\rightarrow\infty}\sup(a_n,a_{n+1},\ldots)$. How can I remember which one is the union of intersections, and which one is the intersection of unions?
Best Answer
We can "identify" each set with its characteristic function
$$\chi_A(x) = \begin{cases}1 &, x \in A\\ 0 &, x \notin A.\end{cases}$$
Then we have
$$\chi_{\liminf A_n}(x) = \liminf \chi_{A_n}(x); \quad \chi_{\limsup A_n}(x) = \limsup \chi_{A_n}(x)$$
for all $x\in X$. The characteristic function of the limes inferior/superior of the sequence of sets is the pointwise limes inferior/superior of the characteristic functions of the sets in the sequence.