Continuing a topic I started a few posts ago, Definite Integrals, I want to discuss some properties of the definition of the integral according to Darbo, and to prove a theorem called Riemann condition.
This post builds heavily on the previous post on this subject, so it will be a good idea to read it before starting to read this post.
In this post f(x) is a bounded function on the interval [a,b] to the real numbers, and P is a division of the interval [a,b].
Firstly, lets show that the infimum of the Upper Darbo Sums (U(P,f)) is alway large or equal to the supremum of the Lower Darbo Sums (L(P,f)). To proof this it is enough to show that for any two divisions P1 and P2 the upper sum according to P2 is greater or equal to the lower sum according to P1. To do this we will need a simple lemma: "If all the points in P are also in P' than
The inequalities are all weak. I will proof one of them and the second one can be proofed exactly the same. Lets suppose that there is only one extra point in P'. We can suppose that this new point is between the first two points in P. Than:
Now we can use this lemma. Lets look on P3 - the union of P1 and P2. According to the lemma:
And this is exactly what I wanted to show.
The next step is to prove the Riemann condition theorem. This theorem says that: "The function f(x) has an integral only and only if:
Since the integral exists we can write than that:
In the next post on this subject I will show (and proof) another condition, which is very similar to Riemann condition but is more easy to work with.