Mikusinski’s Approach to Bochner Integrals – Unconditional Replacement

Question

In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions:

Defn. Let $X$ be a Banach space. A function $f:\mathbb{R}\to X$ is Bochner integrable if there exists a sequence of half-open intervals $[a_i, b_i)$ and a sequence $\lambda_i\in X$, such that

1. The series $\sum (b_i – a_i) \lambda_i$ converges absolutely in $X$; and
2. Whenever $x\in \mathbb{R}$ is such that $\sum \lambda_i \mathbf{1}_{[a_i,b_i)}(x)$ converges absolutely, the series converges to $f(x)$.

My Question: Has someone examined the consequences, if in the definition above, absolute convergence is replaced by unconditional convergence?

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Remarks

When $X$ is finite dimensional, absolute and unconditional convergence are equivalent, so in this case both reduce to the standard Lebesgue integral. So the question is only meaningful when $X$ is infinite dimensional.

Given an arbitrary series $\sum \lambda_i$ in $X$ that is unconditionally convergent but not absolutely convergent, we have that the function $\sum \lambda_i \mathbf{1}_{[i,i+1)}$ is integrable in this unconditional sense, but not absolutely integrable (in the sense that the function $\sum \|\lambda_i\| \mathbf{1}_{[1,i+1)}$ is not in $L^1(\mathbb{R},\mathbb{R})$). So this modified definition is certainly weaker than Bochner integrability.

It is not entirely clear to me if the integral is in fact well-defined with the unconditional convergence: Mikusiński proved that for Bochner integrable functions the integral is independent of the approximating sequence of step functions. His proof however used that such functions are absolutely integrable and the uniqueness of integral holds for real-valued functions. As discussed above, this route is not feasible with only unconditional convergence.

Answer 1

This was studied previously by James Brooks together with Jan Mikusiński; the relevant references are

• Brooks, J. K., Representations of weak and strong integrals in Banach spaces, Proc. Natl. Acad. Sci. USA 63, 266-270 (1969). ZBL0186.20302.
• Brooks, James K.; Mikusinski, Jan, Weak integrals defined on Euclidean (n)-space, Stud. Math. 44, 501-505 (1972). ZBL0242.28006.

They proved:

A function $f:\mathbb{R}^n\to X$, where $X$ is Banach, is Gelfand-Pettis integrable if and only if there exists a countable collection of elements $x_i\in X$ and rectangles $I_i\subset \mathbb{R}^n$ such that

1. $f$ is almost everywhere equal to $\sum x_i \mathbf{1}_{I_i}$, where the series converges absolutely,
2. $\int f = \sum x_i |I_i|$, where the convergence is unconditional.

Compared to the Definition given in my original question, it is important to note that the main result replaces absolute convergence with unconditional convergence only in one of the two conditions.

That in the integration portion, the weak integral and the unconditional coincide is largely due to the Orlicz-Pettis Theorem. I have not yet fully understood why the a.e. pointwise convergence is required to use absolute and not unconditional convergence, but likely it is due to the measurability requirement.