# Where does Markov operator come from?

I found this definition of "Markov operator" in the book Chaos, Fractals, and Noise by Lasota and Mackey.

Denote by $L^1(\mu)$ the space of Lebesgue integrable functions according to the measure $\mu$. Then, an operator $$P: L^1(\mu) \to L^1(\mu)$$ is called Markov Operator when $$f\geq 0 \Rightarrow Pf \geq 0$$ and $$\int (Pf)(x) \mu(dx) = \int f(x) \mu(dx)$$ holds for any $f \in L^1(\mu)$.

But where does this originate from? Who invented it?

• One might think Markov, but no. Kolmogorov. – Conifold Apr 30 '15 at 21:58

This is a continuous analog of (transposed) stochastic matrix, the transition matrix in a Markov chain with discrete set of outcomes. These were introduced in 1906 by Markov apparently to disprove Nekrasov's claim that central limit theorems only applied to independent events, but later found many practical applications. Entries of a stochastic matrix are probabilities covering a complete set of outcomes in each row. So they have to be positive and add up to $1$. A transposed matrix will have positive entries and column sums $1$, which means that it transforms positive vectors into positive vectors, and preserves their entry sums.