For a measure space $(X,\mathcal{A},\mu)$, the space of "observables" with respect to finite set $F$ which is endowed with counting measure on all of its subsets, is defined as follows:

$$obs (X, \mu, F)=\{f:X\to F \,|\, f \,\,\text{is measurable}\}.$$

$obs(X, \mu, F)$ With metric $d$ which measures the points of difference of two functions is a metric space:

$$d(f, g)=\mu \{x|f(x) \neq g(x)\}. $$

My questions are:

1- where the term "observable" comes from and why?

2- Which connections may I expect from this correspondence, by correspondence I mean corresponding a metric space $obs(X, \mu, F)$ to a measure space $(X,\mathcal{A},\mu)$, and which connection exists?

For a measure space $(X,\mathcal{A},\mu)$, the space of "observables" with respect to finite set $F$ which is endowed with counting measure on all of its subsets, is defined as follows:

$$obs (X, \mu, F)=\{f:X\to F \,|\, f \,\,\text{is measurable}\}.$$

$obs(X, \mu, F)$ With metric $d$ which measures the points of difference of two functions is a metric space:

$$d(f, g)=\mu \{x|f(x) \neq g(x)\}. $$

My questions are:

1- where the term "observable" comes from and why?

2- Which connections may I expect from this correspondence, by correspondence I mean corresponding a metric space $obs(X, \mu, F)$ to a measure space $(X,\mathcal{A},\mu)$, and which connection exists?

PS:

• For clearing the question 2, I must explain some more details. $(obs(X, \mu, F), d)$ is a compact metric space (It is compact because pointwise limit of measurable functions is measurable). Now my new question is:
  2': When I must expect for observables space be a geodesic metric space and which geometric properties of this space relates to measure theoretic properties of the space $(X,\mathcal{A},\mu)$?

• As @michael-greinecker 's mentioned, considering the finite set $F$ with counting measure is not important for measurability of observables.