I want to know what information can we infer from multiplying a column vector $f$ by a transition matrix $P$ (of a Markov Chain) on the left.
I was reading a book and they said that:
We might think of $f$ as function on the state space $\chi$.
The $x^{th}$ entry of $Pf$ tells us the expected value of the function $f$ at the next state given that we are at the state $x$.
I didn't understand it quite well.
Consider a Markov chain with transitions given by $P$, with $P_{i,j}=\Pr(X_{t+1}=j\,\vert\, X_t=i)$. Thinking of $f$ as both a function and a vector with $f(j)=f_j$, we have $$ (Pf)_i=\sum_{j=1}^n P_{i,j}f_j=\sum_{j=1}^n \Pr(X_{t+1}=j\,\vert\, X_t=i)f(j)=\mathbb{E}[f(X_{t+1})\vert X_t=i], $$ just using the definition of expectation. This is exactly what you wrote in words.