Introduction to Mathematical Physics/Measure and integration

Lebesgue integral

The theory of the Lebesgue integral is difficult and can not be presented here. However, we propose here to give to the reader an idea of the Lebesgue integral based on its properties. The integration in the Lebesgue sense is a functional that at each element $F$ of a certain functional space (the space of the summable functions) associates a number note $\int f(x)dx$ of $\int f$.

for a function $f$ to be summable, it is sufficient that $|f|$ is summable. if $g(x)\geq 0$ is summable and if $|f(x)|\leq g(x)$ then $f(x)$ is summable and

$|\int f\leq \int |f|\leq \int g$

If $f$ and $g$ are almost everywhere equal, the their sum is egual. if $f(x)\geq 0$ and if $\int f=0$ then $f$ is almost everywhere zero. A bounded function, zero out of a finite interval $(a,b)$ is summable. If $f$ is integrable in the Riemann sense on $(a,b)$ then the sums in the Lebesgue and Rieman sense are equal.