Overview
Increasing and Decreasing Functions
A function {% f %} is strictly less than a function {% g %} if {% f(x) < g(x) %} for any {% x %}. This is denoted as
{% f \ll g %}
A set of functions {% \{ F_1,f_2,...,f_n \} %} is said to be increasing if
{% f_1 \ll f_2 \ll ... \ll f_n %}
Theorem
Given a decreasing set of functions {% f_1 \ll f_2 \ll ... \ll f_n %}
{% \displaystyle inf_i \int f_i d\mu = \int inf_i (f_i) d\mu %}
or written in alternative notation
{% \mu(f_i) \downarrow \mu(f) %}