$$ \newcommand{\sX}{\mathsf{X}} \newcommand{\cX}{\mathcal{X}} $$ The definition of a measure is
A measure on a measurable space $\sX\times\cX$ is a function $$ \mu: \cX\to [0, +\infty) $$ satisfying \begin{align*} \mu(\emptyset) &= 0 && \text{Null Empty Set}\\ \mu\left(\bigcup_i A_i\right) &= \sum_{i} \mu(A_i) && \text{Countably Additive} \end{align*}
The definition of a sigma-finite measure is
Let $\sX \times \cX$ be a measurable space and let $\mu:\cX\to [0, +\infty)$ be a measure on it. We say $\mu$ is a sigma-finite measure if the set $\sX$ is a countable union of measurable sets with finite measure $$ \sX = \bigcup_{n\in\mathbb{N}} A_n \qquad A_n\in \cX \qquad \text{and} \qquad \mu(A_n) < \infty $$
while the definition of a probability measure is
A probability measure $\mu$ on the measurable space $\sX\times \cX$ is a measure $\mu:\cX\to[0, 1]$ with $\mu(X) = 1$
Basically it's a probability measure with total measure $1$.
I was wondering if all probability measures are also sigma-finite. In other words, is sigma-finiteness more general than probability measures?