How Do You Calculate the Lebesgue Integral of e^x from 1 to 10?

[eljose]

let be the Lebesgue integral with a meassure $$\mu$$ then if we call this integral..

$$\int_{X}fd\mu=I$$

my questions are..how would you calculate the integral of f(x)=exp(x) from 1 to 10?..another question let be the lebesgue integral on the interval X=(0,t) would be true that:

$$(\frac{d}{dt}\int_{X}fd\mu=f(t)$$?

thanks...

 

[HallsofIvy]

let be the Lebesgue integral with a meassure $$\mu$$ then if we call this integral..
$$\int_{X}fd\mu=I$$
my questions are..how would you calculate the integral of f(x)=exp(x) from 1 to 10?

The same way you would a Riemann integral. If a function is Riemann integrable on an interval, then it is Lebesque integrable on that interval and the two integrals are the same.

..another question let be the lebesgue integral on the interval X=(0,t) would be true that:
$$(\frac{d}{dt}\int_{X}fd\mu=f(t)$$?
thanks...

Except possibly on a set of measure 0, yes.

 

[mathwonk]

you need some hypotheses of course, but if f is "summable" then it is the derivative a.e. of its indefinite integral. see pages 11 and 47, 48 of Riesz Nagy.