Introduction to Mathematical Physics/Measure and integration
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 of a certain functional space (the space of the summable functions) associates a number note of .
for a function to be summable, it is sufficient that is summable. if is summable and if then is summable and
If and are almost everywhere equal, the their sum is egual. if and if then is almost everywhere zero. A bounded function, zero out of a finite interval is summable. If is integrable in the Riemann sense on then the sums in the Lebesgue and Rieman sense are equal.