Topology on the space of universally integrable functions Let $X$ be a compact space. Let us call a function $f:X\to {\mathbb C}$ universally integrable if it is integrable with respect to each regular Borel measure $\mu$ on $X$ (i.e. a positive functional on ${\mathcal C}(X)$, according to Riesz–Markov–Kakutani representation theorem). Let us denote by ${\mathcal U}(X)$ the space of all universally integrable functions on $X$.
My questions:

  
*
  
*What is known about ${\mathcal U}(X)$?
  
*Is it an algebra (with respect to the pointwise multiplication)? 
  
*Did anybody try to find a natural topology on ${\mathcal U}(X)$?
  

EDIT. This was put on hold as too broad, so I add some details. 


*

*I would be grateful for any references where what I am asking is discussed.

*Nate Eldredge already answered the second of the three questions here, so there is no need to discuss this further. 

*As to the third one, the correct specification will be the following. As I wrote in comments to Eldredge's answer, I believe, there is a topology on ${\mathcal U}(X)$ that turns it into an involutive stereotype algebra (see here, or here, or here) such that the involutive spectrum (see again here) of ${\mathcal U}(X)$ as a set is equal to $X$ (like in the case of ${\mathcal C}(X)$):
$$
\text{Spec}\ {\mathcal U}(X)=X
$$
Of course, such a topology on ${\mathcal U}(X)$ must be much weaker than the $C^*$-topology that Nate Eldredge suggests.
Am I right?
As a hypothesis, the following construction can be discussed: to each regular Borel measure $\mu$ on $X$ one can associate a mapping $\varPhi_\mu:{\mathcal U}(X)\to L(\mu)$. So ${\mathcal U}(X)$ can be naturally embedded into the direct product of the spaces $L(\mu)$.
$$
{\mathcal U}(X)\to \prod_{\mu} L(\mu).
$$
I am now thinking about the topology generated on ${\mathcal U}(X)$ by this embedding (i.e. the initial topology generated by the mappings $\varPhi_\mu$). Is it possible that this is what I need?
 A: Let $M(X)$ denote the set of all regular Borel measures on $X$.  (Note in particular that, by your definition, only finite measures are included.)
For each $\mu \in M(X)$, let $\mathcal{B}_\mu$ denote the completion of the Borel $\sigma$-algebra $\mathcal{B}$ with respect to the measure $\mu$ (that is, $A \in \mathcal{B}_\mu$ iff there exist Borel sets $B_1, B_2$ with $B_1 \subset A \subset B_2$ and $\mu(B_1) = \mu(B_2)$.  It's well known that $\mathcal{B}_\mu$ is a $\sigma$-algebra.  So let $\mathcal{B}_u = \bigcap_{\mu \in M(X)} \mathcal{B}_\mu$, which is also a $\sigma$-algebra.  (In the context of Polish spaces, this is the $\sigma$-algebra of universally measurable sets.) We can then say:

A function $f$ is universally integrable iff it is bounded and $\mathcal{B}_u$-measurable.

Clearly any bounded $\mathcal{B}_u$-measurable function is universally integrable.  Conversely, if $f$ is to be integrable with respect to $\mu$, it must be $\mathcal{B}_\mu$-measurable, so every universally integrable function is $\mathcal{B}_u$-measurable.  Now I claim any universally integrable function must be bounded.  To prove the contrapositive, suppose $f$ is unbounded, so that for each $n$ there exists $x_n \in X$ with $|f_n(x)| \ge 2^n$.  Let $\mu$ be the probability measure which assigns measure $2^{-n}$ to the point $x_n$.  Then $\int |f|\,d\mu = \infty$, so $f$ is not universally integrable.
In particular, $\mathcal{U}(X)$ is an algebra, since a sum or product of bounded $\mathcal{B}_u$-measurable functions is again bounded and $\mathcal{B}_u$-measurable.
A simple and natural topology on $\mathcal{U}(X)$ is the one induced by the uniform norm $\|f\| = \sup_{x \in X} |f(x)|$.  This makes $\mathcal{U}(X)$ into a Banach space, and moreover, a $C^*$ algebra.
A: A pleasant theorem answering first part of your question: "A function is universally integrable if, and only if, it is a uniform limit of step functions.", see "On Universally Integrable Functions" by Solomon Leader,
Proceedings of the American Mathematical Society, Vol. 6, No. 2 (Apr., 1955), pp. 232-234. Available online here: https://www.jstor.org/stable/2032346
