From the book i have been reading:
It seems like i can get somehow the meaning of the expression "assign unit mass to the whole circle" but not exactly enough. Is there any way to think about it more visually? Thanks.
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Do you know what a measure is? (Not essential but it helps.) – Ian Nov 07 '18 at 04:22
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If you are more familiar with classical mechanics, think of a infinitesimally thin ring shaped like the unit circle, with nonuniform mass density given by $K_n(\theta)$. – Nov 07 '18 at 04:45
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@lan i am not sure about what do you want me to get in about "measure". In my experience, i have learnt the measure (in mathematics analysis) like that https://en.wikipedia.org/wiki/Measure_(mathematics). – Minh Nov 07 '18 at 08:51
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@Rahul indeed, i approach the content in the view of studying math, and i dont know much about classical mechanics, thanks anyway. – Minh Nov 07 '18 at 08:55
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If you know what a measure is, $K_n$ are densities of measures, call them $k_n$, with $k_n(A)=\int_A K_n(x) dx$. Your property (a) is just $k_n([-\pi,\pi])=1$. – Ian Nov 08 '18 at 01:35
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I think that we can consider the integral as a sum. So property (a) says that the mean value when taking the sum $K_n(x)$ over the interval $[-\pi,\pi]$ is 1, so as you say, $K_n$ are densities of measures. Indeed, i consider $K_n$ is a thing and divide it into densely infinitely many particle - each has a value mean of 1. And the mass of $K_n$ is so equals $1*(2\pi)=2\pi,$ which equals the integral $K_n(x)$ in the interval. Thanks @lan – Minh Nov 09 '18 at 02:03