Introduction to Mathematical Physics/Measure and integration

Lebesgue integralEdit

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.