# Thread: Proving Integrability

1. ## Proving Integrability

The lecturer proved that The drichlet function is not integrable. However I don't find his proof to be correct(Or I don't fully understand the subject, yet). Because he just said that let there be a Partition P and he defined the points, showed that the upper and lower integrations are different from each other. However he uses the same partition for both. But shouldn't there be other conditions? Such as when the partition consist of the rational points, when it consists of irrational points and when it consists of rational and irrational points. He used the same partition for U and L.

And secondly , the condition of integrability is
L={L(f,P}|P is partition for the closed interval [a,b]}
U={U{f,P}|P is partition for the closed interval [a,b]}

supL<infU (less or equal)

A function is integrable when supL=infU

So how come chosing randomly a partition prove the above?

Thank you.

Note: I uploaded the proof

2. ## Re: Proving Integrability

Originally Posted by davidciprut
he upper and lower integrations are different from each other. However he uses the same partition for both. But shouldn't there be other conditions? Such as when the partition consist of the rational points, when it consists of irrational points and when it consists of rational and irrational points. He used the same partition for U and L.
And secondly , the condition of integrability is
L={L(f,P}|P is partition for the closed interval [a,b]}
U={U{f,P}|P is partition for the closed interval [a,b]}
supL<infU (less or equal)
A function is integrable when supL=infU
So how come chosing randomly a partition prove the above?
Note: I uploaded the proof
The uploaded proof is correct.
Given any partition whatsoever, the $\mathfrak{U}([a,b],P)=b-a~\&~\mathfrak{L}([a,b],P)=0$

3. ## Re: Proving Integrability

Yes now I understand why. Now I understood the definition of mk(xk-xk-1) ... The mk stands for the inf of the values of f(x) in interval xk-xk-1 and there is definitely a irrational there from the density of irrationals and therefore it has to be equal to zero... Therefore L([a,b],P)=0 for every P.