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A. Neumaier

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But most physical observables are not bounded.

This came up in another thread, from which I quote some context:

in Algebraic QFT using C*-algebras, one normally says that observables are self-adjoint bounded operators contained in a region of spacetime. So very strictly speaking AQFT would say that momentum is not an observable. However when I say very strictly, I mean very strictly. AQFT does not say that momentum is not observable, just that no complete measurement of it (enough to specify the point in the spectrum completely) is possible in a finite region. All you can observe in a finite region is operators whose eigenstates are ones whose support in momentum space is finite. The width of these regions is the resolution of the equipment and their number is determined by the highest and lowest momentum states the device can measure.

So momentum is not an observable in a very technical, strict manner. However more truthfully this is just a mathematical way of encoding that you cannot measure momentum "to a point", not that you cannot actually measure momentum.

This happens to be true for momentum but has nothing to do with the problem of boundedness. One component of the electromagnetic field strength at a point x is local (and can in principle be measured arbitrarily well) but is not a bounded variable.

Whereas the projection of a momentum component to a bounded interval is bounded but cannot be measured exactly to the point (only arbitrarily well). But we'd discuss this in a new thread....

Perhaps I'm missing something, please correct me if I am, but the electromagnetic field strength at a point x is not an observable, since it is not an operator, it is only an operator valued distribution.

You are only missing implicitly understood embellishments. In full precision:

The integral of F_12(x) with a real, smooth hat function of narrow support is - by conventional standards - an observable whose support is a bounded region of space-time, but has continuous spectrum, hence is not bounded, and therefore not observable according to your definition.