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In these lecture notes on QM by Simon Rhea & Richie Dadhley based on the first lecture Frederic Schuller, the following is mentioned on page-2:

Recall that in classical mechanics an observable is a map $F : Γ → R$, where $Γ$ is the phase space of the system, typically given by the cotangent space $T^{*}Q$ of some configuration manifold $Q$. The map is taken to be at least continuous with respect to the standard topology on $R$ and an appropriate topology on $Γ$, and hence if $Γ$ is connected, we have $F (Γ) = I ⊆ R$.

What is the basis of wanting the Observable to be continous w.r.t standard topology on $R$ and approproriate topology on $\Gamma$?

tryst with freedom
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  • Hi Tryst with Freedom. Welcome to Phys.SE. Linking to private clouds, dropbox, etc, is for various reasons not acceptable on SE, cf. this meta post. – Qmechanic Apr 09 '23 at 20:11
  • How do you suggest, I post the information here then @Qmechanic – tryst with freedom Apr 09 '23 at 20:17
  • Provide title etc of reference. Link to the official abstract/download page for the lecture notes if possible. – Qmechanic Apr 09 '23 at 20:26
  • I think the request is simply too restrictive. The characteristic function of a set, say a ball, of the phase space is an observable and it is not continuous. Yes-No observables are not continuous, they are just measurable. – Valter Moretti Apr 09 '23 at 20:32
  • @ValterMoretti Fair point. But after consulting some books on classical mechanics again, it seems that this (continuity, among other things) indeed is often postulated/ required. – Tobias Fünke Apr 09 '23 at 22:11
  • Related: https://physics.stackexchange.com/q/1324/2451 and links therein. – Qmechanic Apr 10 '23 at 05:01
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    @Tobias, yes I also found that requirement and it is good if one wants to sticks to classical mechanics. If one wants to pass to QM the relevant structure is the $\sigma$ algebra one of elementary observables and there continuity should be relaxed... – Valter Moretti Apr 10 '23 at 06:48

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If you have small perturbations caused by e.g. the environment of the system, you want that these perturbations in the state only give a correspondingly small perturbation of the values of the observable in question.

If this were not the case, then basically a measurement (of a discontinuous observable) could just be a "random" output of the corresponding measurement device and as such the term "measurement" would be useless, i.e. one could not speak of an observable having a certain value (in some error bound) for a given system...

Tobias Fünke
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