Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion of integral. But Bochner and Lebesgue agree on complex measurable functions. So the question arises:

Is Lebesgue obsolete? Or are there some important aspects one will miss not introducing it?

  • $\begingroup$ How is Lebesgue obsolete? Which $\int$ do you use on the right hand side of the definition? $\endgroup$ – Hagen von Eitzen Sep 22 '14 at 11:45
  • $\begingroup$ @HagenvonEitzen: The basic one for simple functions $\int S\mathrm{d}µ:=\sum_k S_kµ(A_k)$ with $S=\sum_k S_k\chi_{A_k}$. $\endgroup$ – C-Star-W-Star Sep 22 '14 at 11:47
  • $\begingroup$ Your definition of the Bochner integral seems different from the usual one. And $S_n\to F$ in what sense? $\endgroup$ – Harald Hanche-Olsen Sep 22 '14 at 11:49
  • $\begingroup$ @HaraldHanche-Olsen: In fact they're equivalent! That is why I mentioned it as a version. Besides $S_n\to F$ pointwise. $\endgroup$ – C-Star-W-Star Sep 22 '14 at 11:57
  • $\begingroup$ Okay … but the Lebesgue integral is also determined by taking limits of the integral of simple functions. The difference seems mainly technical to me. But I would have to study it more carefully to form a stronger opinion. $\endgroup$ – Harald Hanche-Olsen Sep 22 '14 at 12:15

As an example, suppose that $F : [0,1]\rightarrow X$, where $X$ is a Hilbert space. Suppose that $\|F(t)\|$ is a Lebesgue integrable function. If $(F(t),x)$ is measurable for all $x$, then there is a unique vector--say $\int_{a}^{b}F\,dt$--such that $$ \int_{0}^{1}(F(t),x)\,dt = \left(\int_{0}^{1}F\,dt,x\right),\;\;\; x \in X. $$ This is a very simple integral to define, quite intuitive, powerful, does not require separability, and reduces to the scalar case.

If $X$ is not separable, then I seem to recall that the Bochner integral won't allow you to integrate general such things because $F$ may not be Bochner measurable. But you can definitely see how this might be useful, especially knowing how the integral reduces to scalar cases.

  • $\begingroup$ Ah so your point is that the Lebesgue integral will be useful to integrate those ones that cannot be Bochner integrable due to nonseparable image, is this what you mean? I guess you refer to examples like the $f:[0,1]\to\ell^2([0,1]):t\mapsto\chi_t$. But then the above version could be used to bypass this again as that version allows to generalize to integrate also those ones that cannot be approximated by sequences but nets only. $\endgroup$ – C-Star-W-Star Sep 22 '14 at 12:40
  • $\begingroup$ Hmm, I see there is another problem: When passing from sequences to nets one has to find a new proof for "well-definedness" as the old one won't work anymore - though it doesn't mean that the integral won't be well-defined anymore... $\endgroup$ – C-Star-W-Star Sep 22 '14 at 13:49
  • $\begingroup$ @Freeze_S : It is usually trivial to check conditions for integrability of the integral I gave you, but Bochner measurability is not always so easy to establish, especially if you cannot assume a Schauder basis. Properties of the integral I gave you are easier to figure out, too, and this technique works for any reflexive space. Passing to nets is problematic in any measure theory. I think you about have to use Bochner when $X=L^{\infty}$; that case is messy, but what can you do? $\endgroup$ – DisintegratingByParts Sep 22 '14 at 15:32

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