The question is in the image below. We take a partition to the set of normed values in a linear space. I understand the idea of the valued step function for filtering the values of the partition.
but for:
$$ \int \sigma = \sum_{i}^{N} \Delta t_{i} y_{i} $$
for $ \sigma = (y_{0}, y_{1}, ... , y_{k}) $
I do not understand.
How are the characteristic functions that comprise the step function chosen? Why do they have different intervals and weights? (1, -2, 2, etc)
