Intereting Posts

How to maximize the number of operations in process
Sum and average length of chords
Separability of the space of bounded operators on a Hilbert space
Contour Integration – my solution for real integral is complex?
Maximal gaps in prime factorizations (“wheel factorization”)
Has knot theory led to the development of better knots?
Proving an Inequality Involving Integer Partitions
Finding the catenary curve with given arclength through two given points
For integers $n \neq 0$ is $\sin n$ irrational or transcendental?
Sum of squares diophantine equation
Maximal ideals in polynomial rings with real and complex coefficients
A question concerning measurability of a function
Essay about the art and applications of differential equations?
Counting number of $k$-sequences
New, extremely simple golden ratio construction with two identical circles and line. Is there any prior art?

**Reference**

For a bounded nonexample of integrability see: Riemann Integral: Bounded Nonexample

For a convergence theorem on integral see: Riemann Integral: Uniform Convergence

- Dimension for a closed subspace of $C$.
- Adjoint identity
- Extreme points of unit ball of Banach spaces $\ell_1$, $c_0$, $\ell_\infty$
- Counterexample for the stability of orthogonal projections
- Is $W_0^{1,p}(\Omega)\cap L^\infty(\Omega)$ complete?
- Prove that $C^1()$ with the $C^1$- norm is a Banach Space

For a comparison of integrals see: Uniform Integral vs. Riemann Integral

**Definition**

Given a measure space $\Omega$ and a Banach space $E$.

Consider functions $F:\Omega\to E$.

Denote the measurable subsets of finite mass by:

$$\mathcal{A}_\infty:=\{A:\mu(A)<\infty\}$$

and order them by inclusion:

$$A\leq A’:\iff A\subseteq A’$$

Remember the generalized Riemann integral on finite measure spaces:

$$A\in\mathcal{A}_\infty:\quad\int_AF\mathrm{d}\mu:=\lim_\mathcal{P}\left\{\sum_{a\in A\in\mathcal{P}}F(a)\mu(A)\right\}_\mathcal{P}$$

*(For more details see references above.)*

Define the improper Riemann integral as:

$$\int_\Omega F\mathrm{d}\mu:=\lim_A\left\{\int_AF\mathrm{d}\mu\right\}_{A\in\mathcal{A}_\infty}$$

*(Crucially, this reflects independence of approximation by finite spaces.)*

**Discussion**

For finite measure spaces the improper agrees with the proper as $\Omega\in\mathcal{A}_\infty$.

This way, poles still can’t be handled:

$$\int_0^1\frac{1}{\sqrt{x}}\mathrm{d}x\notin E$$

*(Note that the concept of compact intervals isn’t available in general.)*

For Borel spaces a suitable criterion could be continuity plus absolute integrability:

$$F\in\mathcal{C}(\Omega,E):\quad\int_\Omega\|F\|\mathrm{d}\mu<\infty\implies\int_\Omega F\mathrm{d}\mu\in E$$

How to prove this in the abstract setting?

*(I slightly doubt it…)*

- A question about complement of a closed subspace of a Banach space
- Cardinality of a basis of an infinite-dimensional vector space
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- Fourier Transform calculation
- Is it reasonable to think of the expectation of an infinite-dimensional vector?
- Nontrivial solutions of $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$
- uniqueness of Hahn-Banach extension for convex dual spaces
- Many other solutions of the Cauchy's Functional Equation
- Hamel bases without (too much) axiomatic set theory
- True/False: Self-adjoint compact operator

**Yes, it holds!**

As it is continuous it is Bochner measurable by Pettis’s criterion.

As it is absolutely integrable it is also Bochner integrable.

But it is bounded so on subspaces of finite measure Riemann integrable.

Thus by dominated convergence also improperly Riemann integrable.

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