What is the meaning of the notation $f_{\chi_{[a,b]}}$ where $\chi$ is the charactaristic function of $[a,b]$?

Question

I've seen this notation appear in a handful of texts, particularly on measure theory and lebesgue integration, but I've never seen it explained and Google turns up nothing. Is it simply $f_{\chi_{[a,b]}}(x)=\begin{cases} f(x) &\text{if }x\in [a,b] \\0 &\text{otherwise}\end{cases}$?

Answer 1

COMMENT.- Don't forget that Lebesgue integration uses very general functions and that the Riemann integration generally is based on the consideration of step functions, $g$, concisely written as $$g(x)=\sum_{i=0}^{i=n}c_i\chi_{[a_i,b_i]}(x)$$

This gives you maybe an idea of the use of $f_{\chi_{[a,b]}}(x)=\begin{cases} f(x) &\text{if }x\in [a,b] \\0 &\text{otherwise}\end{cases}$ in which it is subunderstand that this function is just $$F(x)=f(x)\cdot\chi_{[a,b]}(x)$$ which is "a kind of step function" where the non zero parts are not right segments.

(You can see the "analogy" between $g$ and $F$ which is very significant when the $[a_i,b_i]$ intervals tend to be very small).