I am reading section 5.9 on continuous time filtering in Measure Theory and Filtering. Here I have come across a technique, which is used in several of the proofs in the book to show that particular stochastic processes are equal (in some sense).
It goes something like this; Let $\psi$ be a test function. According to the book, that is $\psi \in C^2$ and $\psi$ has compact support. It then seem like if $(X_t(x))_{t \geq 0, x \in \mathbb{R}}$ and $(Y_t(x))_{t \geq 0, x \in \mathbb{R}}$ are stochastic processes (or collections of stochastic processes indexed by $x\in \mathbb{R}$) and if $$ \int_\mathbb{R} \psi(x) X_t(x) dx = \int_\mathbb{R} \psi(x) Y_t(x) dx $$ then we can conclude that $X_t(x) = Y_t(x)$ in some sense for every $t \geq 0$ and $x \in \mathbb{R}$
Does anyone know where I can read about this kind of proof technique using "test functions" in this sense. I am new to distribution theory which as far as I can see also deals with "test functions", but I am not sure if this is the place to go. A reference to a book would be good.