- #1
psie
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- TL;DR Summary
- I'm stuck at something fairly basic I think. Let ##B(r,x)## be an open ball of radius ##r## and center ##x## in ##\mathbb R^n##. It is claimed that ##\chi_{B(r,x)}\to\chi_{B(r_0,x_0)}## pointwise as ##r\to r_0## and ##x\to x_0## on ##\mathbb R^n\setminus S(r_0,x_0)##, where ##S(r_0,x_0)## is the sphere ##\{y:|y-x_0|=r_0\}##. I am stuck showing this.
I'm reading a proof of a lemma that $$A_rf(x)=\frac1{m(B(r,x))}\int_{B(r,x)}f(y)\,dy,$$where ##m## is Lebesgue measure, is jointly continuous in ##r## and ##x## (##A## stands for average). The claim that ##\chi_{B(r,x)}\to\chi_{B(r_0,x_0)}## on ##\mathbb R^n\setminus S(r_0,x_0)## is made in the proof. I think there are two cases to consider. Let ##y\in \mathbb R^n\setminus S(r_0,x_0)##.
- ##|y-x_0|<r_0##, i.e. ##\chi_{B(r_0,x_0)}(y)=1##. Is it then also true that for some sequences ##(x_n),(r_n)## that converge to ##x_0,r_0## respectively, that ##|y-x_n|<r_n## for large enough ##n##? Why? If yes, then ##\chi_{B(r_n,x_n)}(y)=1## for large enough ##n## too.
- Similarly, if ##|y-x_0|>r_0##, is it then true that ##|y-x_n|>r_n##?