1.1 KiB
Math4121 Lecture 35
Continue on Lebesgue Integration
Lebesgue Integration
Definition of Lebesgue Integral
For simple functions \phi = \sum_{i=1}^{n} a_i \chi_{S_i}, given a measure E, the Lebesgue integral is defined as:
\int_{\mathbb{R}^n} \phi \, dm = \sum_{i=1}^{n} a_i m(S_i\cap E)
Given a non-negative measurable function f and a measurable set E.
Define \int_E f \, dm = \sup \left\{ \int_E \phi \, dm : \phi \text{ is a simple function and } \phi \leq f \right\}
(We do allows $\int_E f , dm = \infty$)
For general measurable function f, we can define f^-(x)=\max\{0,-f(x)\}, f^+(x)=\max\{0,f(x)\}. (The positive part of the function and the negative part of the function, both non-negative)
Then f=f^+-f^-.
We say f is integrable if \int_E f^+ \, dm < \infty and \int_E f^- \, dm < \infty. (both finite) If at least one is finite, define
\int_E f \, dm = \int_E f^+ \, dm - \int_E f^- \, dm
We allow for A-\infty = -\infty and A+\infty = \infty for any A\in \mathbb{R}. But not \infty-\infty.