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Lebesgue integral

Definitions

Simple non-negative functions

Given a function of form $\phi = \sum^k_{i=1} \alpha_i \chi_{B_i}$ with $\alpha_i > 0$, the integral is defined to be:

$\int_\mathbb{R} \phi = \sum_{i=1}^k \alpha_i m(B_i)$

where $m(B_i)$ is the Lebesgue measure of $B_i$. We have $\int \phi$ is finite if and only if the Lebesgue measures of $B_i$ are finite.

Non-negative measurable functions

Given a measurable function $f:\mathbb{R} \to [0,\infty]$ the integral is defined to be:

$\int_\mathbb{R} f = \sup\{\int_\mathbb{R} \phi: \phi$ simple, $0 \le \phi \le f \}$

For measurable $E \subseteq \mathbb{R}$, the integral is defined to be:

$\int_E f = \int_\mathbb{R} f.\chi_E$

Measurable functions

For a measurable function $f:\mathbb{R} \to [-\infty,\infty]$, we define functions $f^+, f^-: \mathbb{R} \to [0,\infty]$ to be $f^+ = \max(0,f)$ and $f^- = \max(0,-f)$, then we have $f = f^+ - f^-$ and $|f| = f^+ + f^-$. The integral is defined to be:

$\int f = \int f^+ - \int f^-$

Caution!

The Lebesgue integral is ALWAYS defined for a measurable non-negative function (though it may be infinite). However there are some functions $f$ which are not integrable, and that occurs when $\int f^+ = \infty = \int f^-$. Some functions that are not Lebesgue integrable are Riemann integrable, such as $f(x)=sin(x)/x$ for $x > 1$.

Related theorems

Convergence theorems:

• Monotone convergence theorem (MCT)
• Dominated convergence theorem (DCT)

all these pages adapted with probably insufficient credit from my university's lecture notes