  We introduce a real-valued measure ${m_L}$ on non-Archimedean ordered fields
$(\mathbb{F},<)$ that extend the field of real numbers $(\mathbb{R},<)$. The
definition of ${m_L}$ is inspired by the Loeb measures of hyperreal fields in
the framework of Robinson's analysis with infinitesimals. The real-valued
measure ${m_L}$ turns out to be general enough to obtain a canonical measurable
representative in $\mathbb{F}$ for every Lebesgue measurable subset of
$\mathbb{R}$, moreover, the measure of the two sets is equal. In addition,
$m_L$ it is more expressive than a class of non-Archimedean uniform measures.
We focus on the properties of the real-valued measure in the case where
$\mathbb{F}=\mathcal{R}$, the Levi-Civita field. In particular, we compare
${m_L}$ with the uniform non-Archimedean measure over $\mathcal{R}$ developed
by Shamseddine and Berz, and we prove that the first is infinitesimally close
to the second, whenever the latter is defined. We also define a real-valued
integral for functions on the Levi-Civita field, and we prove that every real
continuous function has an integrable representative in $\mathcal{R}$. Recall
that this result is false for the current non-Archimedean integration over
$\mathcal{R}$. The paper concludes with a discussion on the representation of
the Dirac distribution by pointwise functions on non-Archimedean domains.
