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I need your help finding a good, common notation.

Suppose I have a set of intervals, call them $[a_i, b_i]$ for $i=1,...n$. Now, as I will need those intervals rather frequently, I'm looking for a good notation to write instead of $a_i$ and $b_i$. My main trouble with those is that I'd like to use only one letter for both sides of the interval, as this helps not to clutter the lot of variables I might need elsewhere.

The "base-letter" I would like to use is $\sigma$. What kind of notation would you use? What comes to my mind are the following:

  • $[\sigma^-_i, \sigma^+_i]$
  • $[\sigma_{1,i}, \sigma_{2,i}]$
  • $[\sigma^{(1)}_i, \sigma^{(2)}_i]$
  • $[\overleftarrow{\sigma}_i, \overrightarrow{\sigma}_i]$

All of those I feel are misleading and do not really suite my need. Any idea on a good notation that will not be in my way at further points of my writing? As notation varies greatly depending on area, the text will be a mixture of numerics and statistics? (The $\sigma$, by the way, will not come in the way of standard deviation, as I will hopefully not encounter normal distributions)

Do you fell my idea to use only one letter for both sides is clumsy and I should rather sacrifice another letter?

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Let $K_i$ be a closed bounded interval, so that $K_i = [\min K_i, \max K_i]$, ... –  Zhen Lin May 20 '12 at 17:57
@Zhen Certainly has some appeal and a lot of clarity on what is done. Might be the way to go if no better idea shows up. Thanks! –  Thilo May 20 '12 at 18:06
And if $\min$ and $\max$ seem a little too clumsy, you can always define some notation like $\ell(K_i)$ and $r(K_i)$ for the left and right endpoints. –  Brian M. Scott May 20 '12 at 20:11

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