# 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 $\infty-\infty$.