Lebesgue Integral
Overview
The Lebesgue integral was introduced by Henri Lebesgue. It represents a more general integral than the
Riemann integral,
as it exists for functions that are not Riemann integrable and the integrals agree on
functions that are Riemann integrable.
Preliminaries
The following concepts are utilized in the construction of the
Lebesgue integral.
Limits
Beyond being defined on a wider range of functions than the Riemann integral, the Lebesgue integral is useful in that
limits can be exchanged with the integral in a rather broad set of conditions
{% \lim_{i\to\infty} \int f_i(x) dx = \int \lim_{i\to\infty} f_i(x) dx %}
Lebesgue Integral of Simple functions
Given a measurable space {% \Omega %} with a measure {% \mu %} and a
simple function
{% f : \Omega \rightarrow E %},
the Lebesgue integral {% \int _{\Omega} f d\mu %}
is defined to be
{% \int _{\Omega} f d\mu = \sum_{ci} c_i \mu(f^{-1}(c_i)) %}
Lebesgue Integral
The Lebesgue integral of a function {% f %} (not necessarily simple) is defined as
{% \int _{\Omega} f d\mu = sup[ \int_{\Omega} g d\mu ] %}
where {% g %} is a
simple function and {% f(x) \geq g(x) %}
Topics
