Everyone knows the standard probability interpretation of the quantum mechanics. For example, the wave function of some particle at some time $t$ is $\psi (x,t)$. Therefore, if the particle is detected at the time $t$, the probability density that the detector located at $x$ finds the particle will be $|\psi (x,t)|^2$. By the unitarity, the normalization $\int dx|\psi (x,t)|^2=1$ always holds, whatever $t$ is. Also this probability is irrelevant to the details how the detectors work.
However, if I want to know the probability when the detector will detect the particle, what should I do? To be more specific, I want to evaluate the probability density, denoted by $P(x,t)$, that the particle is detected by the detector located at $x$ and at the time $t$. It should satisfy the normalization $\int dxdt P(x,t)=1$.
According to the conditional probability principles, it should have the form $P(x,t)=f(t)|\psi(x,t)|^2$, for some function $f(t)$. $f(t)$ may be referred to as the probability when a detector anywhere detects the particle. How to evaluate $f(t)$? Does it depend on the details of the detectors? Or it’s just a constant but why?
I used to think this question doesn’t make sense, because the measurement happens once I put a detector, and then the wave function changes dramatically. But later I recognized that the detector is always somewhere in the universe, so it’s hard to talk about when I put this detector.