I'm reading Operator Theoretic Aspects of Ergodic Theory by Eisner et al. In multiple occasions, they use the combination "extension by approximation'' or alike. Specifically, the first part of their Proposition 13.6 reads as follows.
Let $X$ and $Y$ be probability spaces and let $1\leq p \leq \infty$. Then the restriction mapping $\Phi_p:M(X;Y)\to M_p(X;Y)$ with $\Phi_p(S):=S|_{L^p}$ is a bijection.
In the above, $S:L^p(X)\to L^p(Y)$ is a Markov operator so that it satisfies $S \geq 0$, $S1 = 1$ and $\int_Y Sf = \int_X f$ for all $f \in L^p(X)$. $M_p(X;Y)$ is the set of all Markov operators from $L^p(X)$ to $L^p(Y)$ and $M(X;Y)=M_1(X;Y)$.
Then the proof of this part goes as follows.
Each $S\in M_p(X;Y)$ satisfies $||Sf||_1=\int_Y |Sf|\leq \int_Y S|f|=\int_X |f|=||f||_1$ for all $f\in L^p (X)$. Hence, $S$ extends uniquely to a Markov operator on $L^1$ by approximation, and $\Phi_p$ is bijective.
I don't understand what exactly "$S$ extends uniquely to a Markov operator on $L^1$ by approximation" means and I'm hoping you guys can help me out.
The space $L_p (X)$ is dense subspace of $L_1 (X)$ and $S$ is defined on $L_p (X).$ So by Hahn - Banach Theorem you can extend $S$ to an operator $S_1 $ on whole $L_1 (X)$ but since $L_p (X)$ is dense in $L_1 (X)$ the extension must be unique, since every $f\in L_1 (X) $ can be approximate by functions $f_n\in L_p (X) $ such $||f_n -f||_1 \to 0.$