Characteristic functions have magical properties. For example, consider a double summation:

$$\sum_{k=1}^M\sum_{r=1}^k a_{r,k}.$$

How do you switch the order of summation here? A geometric way to think of this is arrange the terms out and “see” that this sum must be equal to

$$\sum_{r=1}^M\sum_{k=r}^M a_{r,k}.$$

I find this unsatisfactory because the whole point of good notation is that you shouldn’t have to think about what it actually means. I do think it’s very important to understand what the notation means, but in doing formal manipulations it’s nice not to have to do that all the time.

A superior proof that these two sums are equal would be to write

$$\sum_{k=1}^M\sum_{r=1}^k a_{r,k} = \sum_{k=1}^M\sum_{r=1}^M a_{r,k}\delta(r\leq k)$$

where $\delta(r\leq k)$ is the function of the variables $r$ and $k$ that equals one exactly when $r\leq k$ and zero otherwise. Then we can happily switch the order of summation to get

$$\sum_{r=1}^M\sum_{k=1}^M a_{r,k}\delta(r\leq k).$$

Now, it’s trivial to get rid of the $\delta(r\leq k)$ function by writing

$$\sum_{r=1}^M\sum_{k=r}^M a_{r,k}.$$