The following is a wonderful property of the Lebesgue Integral, also known as absolute continuity of Lebesgue Integral. Basically, it means that whenever the domain of integration has small enough measure, then the integral will be arbitrarily small.
Suppose is integrable.
Given , there exists such that for all measurable sets with , .
Define for . Each is measurable and . Note that
Let . Then is a sequence of non-negative functions such that . By Monotone Convergence Theorem, , that is,
Let be sufficiently large such that .
Let , and suppose . Then