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$I$ : finite index set, $f_i$ is LSC for each $i \in I$

I wanna prove below. $$\min_i f_i \text{ is LSC}$$

And is there some example of that does not extend to infinite $I$?

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NICE HINT, and good example and proof attached. Thanks! –  japee Jun 10 '12 at 9:36
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2 Answers 2

up vote 3 down vote accepted

We use properties of $\liminf$. First, it's enough to show the result when $I$ has two elements (then use induction on the number of elements of $I$). If $f$ and $g$ are lower semi-continuous, and $x_0\in\Bbb R$, if $x_n\to x_0$ then \begin{align*} \liminf_{n\to +\infty}\min\{f(x_n),g(x_n)\}&=\lim_{k\to +\infty}\inf_{n\geq k}\min\{f(x_n),g(x_n)\}\\ &=\lim_{k\to +\infty}\min\{\inf_{n\geq k}f(x_n),\inf_{n\geq k}g(x_n)\}\\ &=\min\{\liminf_{n\to +\infty}f(x_n),\liminf_{n\to +\infty}g(x_n)\}\\ &\geq \min\{f(x),g(x)\}. \end{align*} Since it's true for all sequence you can conclude lower semi-continuity.

When $I$ is infinite, it's not true any more. Take $f_i$ the indicator function of $(-\infty,1/i)$. Since this interval is open, $f_i$ is LSC. We have that $\min_{i\in\Bbb N}f_i=\chi_{(-\infty,0]}$, which is not LSC (take $x_0=0$ and $x_n=1/n$).

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HINT: A function $f:\Bbb R\to\Bbb R$ is lower semicontinuous if and only if $\{x\in\Bbb R:f(x)>\alpha\}$ is open for each $\alpha\in\Bbb R$. (If you’ve not yet seen this characterization, you should try to prove it from whatever definition of lower semicontinuity you have; it’s quite useful.)

For any functions $f,g:\Bbb R\to\Bbb R$ and any $x,\alpha\in\Bbb R$, $\min\{f(x),g(x)\}>\alpha$ if and only if $f(x)>\alpha$ and $g(x)>\alpha$, so $$\Big\{x\in\Bbb R:\min\{f(x),g(x)\}>\alpha\Big\}=\{x\in\Bbb R:f(x)>\alpha\}\cap\{x\in\Bbb R:g(x)>\alpha\}\;.$$ Since the intersection of any finite number of open sets is open, the result follows immediately.

The intersection of an infinite collection of open sets need not be open, so you should expect that there is an infinite family of lower semicontinuous functions whose minimum is not lower semicontinuous. Try using the indicator functions (also called characteristic functions) of a nested family of open intervals whose intersection is not open.

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