# How to understand observables in quantum field theory

I am reading a paper about quantum field theory, something that I am new to. I have some experience with quantum mechanics. In the paper, it explains how a field is a function from a spacetime manifold(M) to a vector space(V):

Φ:M→V

The paper then explains the form of the observables. It explains how an observable is the value of Φ at x∈M, but our calculations will work out better if we instead look at average values of Φ in small regions of space, as infinitely much energy is required to know the value of a field at a point. To calculate the average values of the field in a small region of space, and thus our observable, the paper defines a Schwartz space, S, as a space of Schwartz functions of the form f:M→V*. It then explains that the observables are of the form:

$$O_f (Φ) = \int_M 〈f,Φ〉$$

So at this point I am confused. How does this integral lead to an average value of the field within a region of space? I would really appreciate any help showing me how the integral above is indeed the form of an observable for the average value of the field within a small region in space. Specifically, how does one carry out the calculation that gives them a value for $$O_f$$? Thanks in advance for any help.

The following is a Schwartz function, for any $$\epsilon > 0$$:
$$f_\epsilon (x) = N \int e^{-(\|\vec{x}-\vec{y}\|^2 + (x^0 -y^0)^2)/\epsilon^2} \theta(\|\vec{y}\|-R) \theta(|y^0|-T/2) d^4 y$$
This is a smoothened version of the function which is $$1$$ in a ball of radius $$R$$ in space and for a time duration $$T$$, and zero outside of that. Integrating an observable against $$f_\epsilon(x)$$ thus, heuristically, gives the average of the observable in the ball of radius $$R$$, in a time interval of length $$T$$.