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Suppose that we have an interval $[a,b]$. Define a semi-partition of $[a,b]$ to be any ordered list of points $x_1 < ... < x_n$ all contained in $[a,b]$ ($a$ and $b$ need not be included).

Given two semi-partitions $X = {x_1, ..., x_n}$ and $Y = {y_1, ..., y_m}$ it is clear that they can be combined to make a denser, refined semi-partition. I am wondering if anyone knows a good way to combine $X$ and $Y$ to make a new semi-partition $Z$ such that the density of points in $Z$ is roughly the average of the densities of points in $X$ and $Y$.

I think my main problem is, I can't think of any good way to define the 'density of points' for a semi-partition. If anyone has any ideas I would really appreciate it! Thank you

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Why can't you just define density as $|X| / (b-a)$? If it's an ordered list it has measure zero, so if infinite then $X\cup Y$ has measure zero as well (i.e. the "same" density). –  Xodarap Oct 5 '12 at 17:14
@Xodarap: That's an ok definition of density, but it doesn't take into account anything about where the points in X are. I was trying to think of density as a function on [a,b], and also trying to think how can one go back and forth between such a function and a semi-partition. My end-goal is to come up with a good density function, and then a good way to average two density functions to create a new semi-partition. –  Eric Haengel Oct 8 '12 at 15:52

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