Elements of multiple riemann integration

[Castilla]

Hello guys, I am following the chapter about multiple Riemann integrals in Apostol's Mathematical Analysis. Theorem 14.11 says this (I translate from spanish to english):

"Let S be a Jordan measurable set. Let the function f be defined and bounded in S. Then f is Riemann integrable if and only if the discontinuities of f in S are a set of cero measure".

The proof says:

Let L be a compact interval that contains S and let g(x) = f(x) when x belongs to S, g(x) = 0 when x belongs to L - S. The discontinuities of the function f will be discontinuities of the function g. But the function g may have also discontinuities en some or all the points of the boundary of S (...).

Now, I understand that there are only two logical options:

- S includes the boundary of S.
- S does not includes the boundary of S.

Now, if S includes the boundary of S then the function f is defined and bounded in the boundary of f and therefore the only discontinuities of g in L are those of f in S.

And if S does not includes the boundary of S, L - S includes it. Then if x is a point in said boundary g(x) is always 0, then g is constant in said boundary, hence g is continuous in said boundary. And so we have the same result that in the other case: the only discontinuities of g in L are those of f in S.

But if this is right, why the proof says that g may also have discontinuities in some or all the points of the boundary of S? Where it is my flaw??

Maybe this is a kind of stupid question. If it is so, I beg your pardon for wasting your time. Nevertheless I dare to ask your help.

Thanks in any case,

Pedro Castilla
Lima, Perú

 

[Hurkyl]

For any point in the domain of a function, the function is continuous in {P}. But, it might not be continuous at P...

 

[mathwonk]

just off the top of my head here, define the "oscillation" of a function f at a point p to be the glb of the diameter of the images under f of all open nbhds of p. I.e. if f is consinuous at p, then as we take smaller interval nbhds of p, their images under f have diameters going to zero. If not continuous at p, the diameters of the images of intervals like (p-1/n,p+1/n) have a positive lower bound, the oscillation.

Now if the set of discontinuities has positive measure, i.e. the set of points with positive oscillation, has positive measure, then the union over all n, of the sets with oscillation at least 1/n, has positive measure, so there is a set of positive measure (say a>0) with oscillation 1/n for some fixed n.


Then you are in trouble with the riemann integral, since in any subdivision of the domain interval, the sum of the lengths of those intervals containing such a point is at elast 1/n, hence the upper and lower integrals for that subdivison differe by at elast a/n. since this hoilds for all subdivisions, the upper and lwoer integrals do not have the same limit.

now i suppose one can prove the converse equally easily.

 

[Castilla]

Hurkyl, please put the idea a little bit clearer.

 

[mathwonk]

have I not fairly completely settled this matter?

 

[Castilla]

Mathwonk:

You are sketching the proof of this: the function f is Riemann-integrable in (say) set A if and only if the set of its discontinuities in A has cero measure.

That I understand, but my impasse is other.

The theorem in Apostol's book (14.11) says: Let S be a Jordan measurable set. Let the function f be defined and bounded in S. Then the function f is Riemann integrable if and only if the discontinuities of f in S are a set of cero measure".

Apostol's proof begins this way:

"Let L be a compact interval that contains S.

Let be a function g so that g(x,y) = f(x,y) when (x,y) belongs to S and g(x,y) = 0 when (x,y) belongs to L - S.

The discontinuities of the function f will be discontinuities of the function g. BUT THE FUNCTION G MAY HAVE ALSO DISCONTINUITIES IN SOME OR ALL THE POINTS OF THE BOUNDARY OF S (...)."

That's the point I am not so sure of being understanding. Let me explain why.

We can have any of these (I am amending a slight mistake of my original thread):

- S includes Bd S. (1).
- S does not includes Bd S. (2).
- S includes some points of Bd S, and L - S includes the other points of Bd S. (3).


(1) S includes Bd S. So we know that in S U Bd S the function g is equal to the function f. In any other point of L we have g(x,y) = 0. So the only discontinuities of the function g in L are those of f in S.

(2) S does not includes Bd S. Then L - S includes Bd S and we know that in L - S the function g is defined g(x,y) = 0. So -again- the only discontinuities of the function g in L are those of f in S.

(3). This is a mixture of (1) and (2) and the conclusion is the same: the only discontinuties of the function g in L are those of f in S.

Then, when in his proof Apostol says that the function g may also have discontinuities in some or all the points of the boundary of S, is he only referring to the posibility that some (or all) points of Bd S belonging to the domain of the function f ??

Thanks.

 

[mathwonk]

i don't get it. if you understand the proof that a function is riemann integrable iff its discontinuites have measure zero, why do you care what apostol says about it? i certainly don't.