Can the Lebesgue Integral of a_p be Defined for Non-Differentiable Functions?

[eljose]

Let define the function:

$$a_{p}(x)= 1$$ if x is an integer and prime and 0 elsewhere, my

question is...what would be its Lebesgue integral let,s say from [c,d] with c and d positive and real..

 

[HallsofIvy]

Since this function is only non-zero on only a finite number of points between c and d, isn't the integral obviously 0?

 

[matt grime]

It's Riemann (and hence lebesgue) integral is rather trivially zero on any interval. My question is why would you need to ask this?

 

[eljose]

then why the integral of the function $$f(x)=1$$ iff x is rational and 0 elsewhere is different from 0?...

 

[shmoe]

then why the integral of the function $$f(x)=1$$ iff x is rational and 0 elsewhere is different from 0?...

You're sayig the Lebesgue integral of this function is non zero? How do you figure?

 

[matt grime]

1 on a set of measure zero 0 every where else, aka almost everywhere zero. that the integral is zero of such a thing is practically the point of lebesgue theory.

 

[HallsofIvy]

then why the integral of the function $$f(x)=1$$ iff x is rational and 0 elsewhere is different from 0?...

It isn't! Who told you that it was? The lebesque integral of the function you give is 0 over any finite interval.

IF, instead, you define f(x)= 1 if x is irrational and 0 if x is rational (1- your f(x)) then the integral of f over the interval [a, b] is b-a.

 

[eljose]

i know i have posted this topic or analogue before but i have the doubts with lebesgue integration:

a) the Lebesgue integral of exp(x)..is equal to Riemann integral of exp(x)

b) $$D_{t}\int_{0}^{t}d\mu{f}= f ?$$

c)what would be the formula for integration by parts in Lebesgue integration?..

thanks.

 

[matt grime]

integration by parts requires the integrands to be differentiable or to be a derivative and hence continuous, so there is no point in using lebesgue integration, is there?