Intereting Posts

Obtain magnitude of square-rooted complex number
Show from the axioms: Addition in a quasifield is abelian
subgroup generated by two subgroups
Schröder-Bernstein for abelian groups with direct summands
Proving $2^{2n}-1$ is divisible by $3$ for $n\ge 1$
Find $g'(x)$ at $x=0$
How find that $\left(\frac{x}{1-x^2}+\frac{3x^3}{1-x^6}+\frac{5x^5}{1-x^{10}}+\frac{7x^7}{1-x^{14}}+\cdots\right)^2=\sum_{i=0}^{\infty}a_{i}x^i$
Finitely but not countably additive set function
Weird and difficult integral: $\sqrt{1+\frac{1}{3x}} \, dx$
How to differentiate a homomorphism between two Lie groups
Evaluate $\int _{ 0 }^{ 1 }{ \left( { x }^{ 5 }+{ x }^{ 4 }+{ x }^{ 2 } \right) \sqrt { 4{ x }^{ 3 }+5{ x }^{ 2 }+10 } \; dx } $
Prime number theory and primes in a specific interval
Quotient geometries known in popular culture, such as “flat torus = Asteroids video game”
Period of a sequence defined by its preceding term
Fixed Set Property?

**Reference**

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

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

- Hilbert Space is reflexive
- How to prove Campanato space is a Banach space
- Inclusion of $\mathbb{L}^p$ spaces, reloaded
- Proof of Pitt's theorem
- Is duality an exact functor on Banach spaces or Hilbert spaces?
- If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?

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…)*

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- Trace operators on topological vector spaces
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**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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