Re: Riemann sums + integrability

On 2007-11-07 02:56:06 -0500, Robert Israel <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> said:

ame <raelien@xxxxxxxxx> writes:

i've just learned that the definite integral of f is equal to the
limit of the riemann midpoint sum if f is continuous in the interval.
i'm curious -- can that limit exist for a function that's not riemann
integrable? if so, what would such a function be? if not, why not?

Yes it can exist. Let's say your interval is [0,1]. Your Riemann midpoint
sums all involve evaluating f at rational numbers. Consider a function
that is 0 on the rationals (so all the Riemann midpoint sums are 0), but
does something wild and crazy on the irrationals...

You seem to be considering only nets of partitions with points in the rationals only. In that case, all midpoints are rationals also. However, what if you consider also nets of partitioners with points in all of reals too?

For a partition with points in the irrationals, the midpoints need not be rationals. So, the limit of riemann midpoint sum will not converge in your example. That is, I can always find a finer partition with irrational points such that the midpoints are also irrational.

Did I misunderstood something here?

--

-kira

.