Random signals as power signals
Note that the condition

$$\int_{-\infty}^{\infty}|f(t)|^2dt&lt;\infty\tag{1}$$

(i.e., that the signal $f(t)$ has finite energy) is very restrictive when we try to model signals, even though obviously any actually occurring signal must have finite energy. Modeling signals as random processes means that we ignore condition $(1)$. Models are always unrealistic to a certain degree, but many signals can be described very well by random processes even though the signals have finite energy and their models do not. This aspect of the model is often irrelevant.

One example which may serve to understand this fact a bit better is the frequently used model of a (wide-sense) stationary process. Certain statistical properties of such a process do not change over time, and, consequently, realizations of such a process will generally not decay as $t\rightarrow\pm\infty$, and $(1)$ will generally not be satisfied, even though we are only interested in the properties of that process during a certain finite time window. However, power and the power spectrum can be defined for such processes, and most practically useful processes have finite power (or can easily be made to have finite power).