Point evaluation is represented by a Dirac measure

Evaluation is linear, and $f\ge 0$ implies $f(x)\ge 0$, so $L_x$ is positive.

For a nonnegative simple function $s$, the definition of its integral [given] and the set formula for ${\delta }_x$ give $\int sd{\delta }_x=s(x)$. Increasing simple approximation and monotone convergence extend this identity to every nonnegative measurable function, and positive/negative parts extend it to every integrable real function. In particular, ${\int }_{X}fd{\delta }_x=f(x)=L_x(f)$ for every $f\in C_c(X)$.

The measure ${\delta }_x$ is finite on compact sets. If a Borel set $E$ [step 1.2] contains $x$, every open superset has ${\delta }_x$-measure one; if it does not, the open set $X\setminus \{x\}$ contains $E$ and has measure zero. Thus ${\delta }_x$ is outer regular. Likewise an open set containing $x$ contains the compact set $\{x\}$, while the empty compact set suffices otherwise, so ${\delta }_x$ is inner regular on opens. Hence ${\delta }_x$ is a Radon measure representing $L_x$ by step 1.2.