On the definition of locally integrable functions on an abstract measure space.

Question

Recently I've been able to find a very cheap copy of the nice monograph [1], where I can find (chapter 10, §1, p. 163) the following general definition of Locally integrable function (respect to a measure $\mu$)

Let $\mathscr C$ be a clan of subsets of $T$, $E$ a Banach space and $\mu$ a positive measure on $\mathscr C$.
Definition 1. A function $\boldsymbol f$ defined on $T$ with values in $E$ or in $\overline{\Bbb R}$ is said to be locally integrable with respect to $\mu$, or locally $\mu$-integrable, if for every $A\in\mathscr{C}$ the set function ${\boldsymbol f}\varphi_A$ is $\mu$-integrable.

In the above definition

• A clan $\mathscr C$ ([1], chapter 1, §1, p.1, also called a ring of sets as in [2], chapter I, §4, p. 19) is a family of subsets of $T$ (i.e. $\mathscr C\subset\mathscr P(T)$) characterized by the following two properties

  1. $A\setminus B\in\mathscr{C}$ for all $A, B\in \mathscr{C}$.
  2. $A\cup B\in\mathscr{C}$ for all $A, B\in \mathscr{C}$.

• $\varphi_A$ is the characteristic function of $A\in \mathscr C$, i.e. the function $$ \varphi_A (x)= \begin{cases} 1 & x\in A\\ 0 & x\notin A \end{cases} \qquad A\in\mathscr C. $$

Question. How can I show that, when $T$ is a finite dimensional vector space, or more generally a topological vector space, definition 1 above reduces to the standard one(s) as given for example in the Wikipedia entry "Locally integrable function"?

What I've tried. Basing on the above definition of a clan/ring of sets, I tried to see if the properties of these families of sets imply their compactness when defined on a topological space, but I failed. Precisely, I was not able to prove that for any given open covering $\mathscr{B}$ of the sets $A\in\mathscr{C}$ it is possible to select one $\mathscr{B}_f\subseteq\mathscr{B}$ containing finitely many members.

Context. Years ago I contributed a lot to the said Wikipedia entry: jointly with other contributors we were able to describe quite accurately the case of locally integrable functions on a finite dimensional measurable space $T$, i.e. $T=\Bbb R^n$, $n\ge1$. Moreover, as stated in the entry itself, we also stated that the extension of the concept when $T$ is a general topological space does not involve any true difficulty, since the meaning of compactness of a set $A\subset T$ is obvious as in the finite dimensional case. Nevertheless I know that the concept of a locally integrable function makes sense also when $T$ is simply an abstract set without any other assumptions: since a description of this extension is also asked by an anonymous commenter in the entry talk page, I aim to complete the entry in that sense.

References

[1] Nicolae Dinculeanu, Vector measures (English), Hochschulbücher für Mathematik. 64, Berlin: VEB Deutscher Verlag der Wissenschaften, pp. X+432 (1966), MR206189, Zbl 0142.10502. (Also published by Pergamon Press in 1967 as volume 95 of their "International Series of Monographs in Pure and Applied Mathematics").

[2] Paul R. Halmos, Measure theory, 2nd printing, (English) Graduate Texts in Mathematics, 18, New York-Heidelberg-Berlin: Springer-Verlag, pp. XI+304 (1974), MR0033869, Zbl 0283.28001.

Answer 1

Summary of the discussion in comments and simplification.

Compact sets form the set $\mathscr{K} \subseteq \mathscr{P}(T)$ the with property:

1. $A\cup B\in\mathscr{K},$ given $A,B \in\mathscr{K}$

Construct a minimal clan $\mathscr{C} \supseteq \mathscr{K}$ starting with elements of $\mathscr{K}$ and applying finite number of steps:

1. $A\cup B\in\mathscr{C},$ given $A,B \in\mathscr{C}$
2. $A\setminus B\in\mathscr{C},$ given $A,B \in\mathscr{C}$

If $\boldsymbol f$ is locally integrable wrt $\mathscr{C},$ then it is trivially locally integrable wrt $\mathscr{K}.$

$\mu$ is Radon measure. This is the case for Lebesgue measure. Under this assumption:

1. Elements of $\mathscr{C}$ are $\mu$-integrable. For $\phi_K, K\in\mathscr{K}$ this is the property of Radon measure.
2. If $\boldsymbol f$ is locally integrable wrt $\mathscr{K},$ then it is locally integrable wrt $\mathscr{C}:$

1. take $C\in\mathscr{C},$ it is constructed in finite steps from elements of $\mathscr{K}.$ If one step was used, done.
2. Otherwise, $C=C'\cup C'',$ or $C=C'\setminus C''$ and constructions of $C', C''\in\mathscr{C}$ involve fewer steps, use induction hypothesis and formulas: $$\phi_{C\cup C'} = \phi_{C} +\phi_{C'} -\phi_{C}\phi_{C'},$$ $$\phi_{C\setminus C'} = \phi_{C} -\phi_{C}\phi_{C'}$$ "$|\boldsymbol f\phi_C|, |\boldsymbol f\phi_{C'}|$ are $\mu$-integrable ($\mu$-measurable + upper integral is finite)" implies $$|\boldsymbol f\phi_C\phi_{C'}| = \sqrt{|\boldsymbol f\phi_C|}\sqrt{|\boldsymbol f\phi_{C'}|} = |\boldsymbol f\phi_C||\phi_{C'}| $$ is $\mu$-measurable (product of measurable functions, first equality) and upper integral is finite (second equality).

Additional comments:

1. $\mathscr{C}$ consists of relatively compact sets, trivially;
2. Under the same assumption about $\mu,$ one can construct another equivalent clan containing only relatively compact sets: $$ \mathscr{C}_1 = \{C\in \mathscr{P}(T): \phi_C \text{ is } \mu\text{-measurable and } C\subseteq K, \text{ for some }K\in\mathscr{K}\} $$
3. Locally integrable function in the case of Lebesgue measure, assumed that $\boldsymbol f$ is measurable from the start. This is redundant if one assumes $\boldsymbol f \phi_K$ is measurable for each $K\in \mathscr{K}:$ these function are even integrable and local measurability is equivalent to measurability.
4. I wanted to be on safer side and assumed that $\mu$ is Radon. I think this assumption can be dropped. Claim "$C\in \mathscr{C}$ are integrable" and claim about $\mathscr{C}_1$ become false.

Answer 2

here is my try i think that that if we want to show the given definition of locally integrable functions with respect to a measure $\mu$ reduces to the standard definition in the case of a finite-dimensional vector space or a more general topological vector space, we need to carefully analyze the conditions and assumptions involved in both definitions.

Let's denote the standard definition of locally integrable functions on a topological space $T$ with respect to a measure $\mu$ as Definition 2. For the purpose of clarity, let's state Definition 2:

Definition 2 (Standard Definition): A function $f$ on a topological space $T$ is said to be locally integrable with respect to a measure $\mu$ if, for every compact subset $K$ of $T$, there exists a $\mu$-integrable function $g_K$ such that $f$ coincides with $g_K$ almost everywhere on $K$.

Now, let's establish the connection between Definition 1 and Definition 2 in the context of a finite-dimensional vector space or a topological vector space.

1. Finite-Dimensional Vector Space:

   • In a finite-dimensional vector space, every closed bounded subset is compact. This is a consequence of the Heine-Borel theorem, which states that in Euclidean spaces, a subset is compact if and only if it is closed and bounded.
   • Considering that every subset is compact, Definition 1, which involves the local integrability over sets in a clan $\mathscr C$, is essentially equivalent to requiring integrability over every compact subset.
   • Therefore, in the finite-dimensional case, Definition 1 reduces to Definition 2, as every set in the clan is compact.

2. Topological Vector Space:

   • In a more general topological vector space, we need to consider the topology of the space and the properties of the clan $\mathscr C$.
   • Definition 1 involves the clan $\mathscr C$, which needs to satisfy certain properties. If $\mathscr C$ is chosen in such a way that it includes compact sets or satisfies conditions equivalent to local compactness, then Definition 1 reduces to Definition 2.
   • The topology of the topological vector space plays a crucial role in determining the properties of the clan $\mathscr C$ that would lead to equivalence with Definition 2.

when $T$ is a finite-dimensional vector space, or when the topological vector space has certain properties such as local compactness, Definition 1 reduces to the standard definition of locally integrable functions (Definition 2). The key lies in understanding the relationship between the clan $\mathscr C$ and the compact subsets of the space, which ensures that local integrability over sets in $\mathscr C$ is equivalent to local integrability over compact subsets.

Answer 3

The way the notion of locally integrable function is given in the OP seems to depend on the ring (clan); also, it seems to be too broad. For example, consider $(\mathbb{R},\overline{\mathcal{B}}^\lambda(\mathbb{R}),\lambda)$, where $\lambda$ is Lebesgue's measure on the real line and $\overline{\mathcal{B}}^\lambda(\mathbb{R})$ is the completion of the Borel $\sigma$-algebra in $\mathbb{R}$ with respect to $\lambda$. The family $\mathcal{C}_f$ of finite subsets $\mathcal{R}$ is a ring (clan) of measurable set, and according to the definition in the OP, any real-extended valued function on $\mathbb{R}$ is locally integrable. The space of all real-extended functions is not very useful or interest from the point of view of Lebesgue integration (or Bochner's itegration) however. From this one sees that:

• Some rings are more useful (or meaningful) than others, and that one should talk about $\mathcal{C}$-locally integrable functions in $(T,\mathcal{F},\mu)$, where $\mathcal{C}$ is a ring.
• To yield an interesting family of locally integrable functions, a ring should have additional properties related to the underlying space $(T,\mathcal{F},\mu)$. The minimal requirements should probably be that $\mathcal{C}\subset \mathcal{F}$ and that $\mu(C)<\infty$ for all $C\in\mathcal{C}$. Still, these simple properties are too broad.

For example, in a locally compact Hausdorff space $T$ equipped with the Borel $\sigma$-algebra, the set $\mathcal{K}_{rc}$ of all relatively compact sets form a ring. Furthermore, due to the Riesz-representation theorem (if one follows the ways of Lebesgue-Charatheodory) or the Daniell-Stone theorem (following the ways of Daniell), the ring $\mathcal{K}_{rc}$ is a determining class for the collection of Radon measures (measures associated to linear functionals on $\mathcal{C}_c(T)$), that is if two Radon measures coincide on $\mathcal{K}_{rc}$, then the measures are the same.