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I'm considering a set valued mapping $X(t): \mathbb{R} \to \mathcal{P}(\mathbb{R}^n)$, where $\mathcal{P}$ denotes the power set. Given a paramater $t \downarrow 0$, I thought I could define the $\underset{t \downarrow 0}{\limsup} X(t)$ as

$$\underset{t \downarrow 0}{\limsup} X(t) := \lim_{t \downarrow 0}\big ( \sup \{x \in \mathbb{R}^n \ : \ x \ \text{is a limit point of } X(t) \} \big )$$

This, however, appears to be wrong. I am told that the limit should look something like $\underset{t \downarrow 0}{\limsup} X(t) = X \subset \mathbb{R}^n$ and I believe my definition gives a single limit point instead. If anyone more familiar with these definitions could help me out I would really appreciate it!

Lgate8
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  • Often, indicator functions are helpful in this context. You probably want to look at something like the lim sup of the sequence of indicator functions giving the indicator function of the limit. – Paul Jul 17 '19 at 18:43
  • do you have some kind of reference as to how that is done? – Lgate8 Jul 17 '19 at 18:49

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Besides indicator functions, you can define a lim sup to be $$\limsup_{t \downarrow 0} X(t)=\{x \in \mathbb{R}^n: \forall t \in \mathbb{R}^+, \exists t_1 \in (0,t) s.t. x \in X(t_1)\} $$

Paul
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  • is that equivalent to $\limsup_{t_k \downarrow 0}X(t_k) := \bigcap_{k = 0}^{\infty} \bigg(\bigcup_{m = k}^{\infty} X(t_m) \bigg) $ ? – Lgate8 Jul 17 '19 at 19:11
  • It is equivalent as long as there are only a countable number of values that $t$ takes. The one I gave works whether or not this number is countable or not. – Paul Jul 17 '19 at 19:14
  • okay, I see the difference. Thanks so much for the help! – Lgate8 Jul 17 '19 at 19:16