  Riesz representation theorem, Daniell-Stone theorem for Daniell integrals and
Stone's representation theorem for probability and measure algebras are three
important classical results in analysis concerning existence of measures with
certain properties. Many proofs of these theorems can be found in the
literature of analysis, from elementary ones which use ordinary techniques from
measure theory, to more sophisticated ones, such as those employing techniques
from nonstandard analysis, in particular for Riesz representation theorem. In
this paper, as the first goal, we give new proofs for all these three theorems.
Our proofs have a mild logical flavor and are uniform in the sense that they
are all based on the same general idea and rely on the application of the same
technical tool from logic to measure theory, namely logical compactness
theorem. In fact, as the second goal of the paper, we try to reveal more the
power of logical methods in analysis in particular measure theory, and make
stronger connections between analysis and logic. We use the setting of
"integration logic" which is a logical framework (and one of the forms of
probability logics) for studying measure and probability structures by logical
means. Indeed, we elaborate this setting and use its expressive power and a
version of compactness theorem holding in it to show its application in measure
theory by giving new proofs for the above-mentioned measure existence theorems.
As mentioned, an advantage of these proofs is that they are all given in a
uniform way since they are all based on the logical compactness theorem. The
paper is mostly written for general mathematicians, in particular the people
active in analysis or logic as the main audience. So it is self-contained and
the reader does not need to have any advanced prerequisite knowledge from logic
or measure theory.
