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.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
{% \displaystyle \lim_{i\to\infty} \int f_i(x) dx = \int \lim_{i\to\infty} f_i(x) dx %}
(see krantz preface)
Preliminaries
The following concepts are utilized in the construction of the Lebesgue integral.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
{% \displaystyle \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
{% \displaystyle \int _{\Omega} f d\mu = sup[ \int_{\Omega} g d\mu ] %}
where {% g %} is a simple function and {% f(x) \geq g(x) %}