The reduced power and reduced product constructions are generalizations of the ultrapower construction that use filters instead of ultrafilters, and you can construct a filter from a measure(and hence a reduced power) simply by taking the collection of all sets whose complement is measure 0. See the book "A Course in Universal Algebra" at http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf for the definition of a reduced power.

There is a weak version of Łoś's theorem that holds for reduced powers although it is a bit messier for reduced powers which we shall state here. This result basically gives a method of determining the truth value of a sentence in a reduced power. In the book "Model Theory" by Chang and Keisler section 6.3, the author describes an effective procedure that does the following. Given any first order sentence $\phi(x_{1},...,x_{n})$ in some first order language $\mathcal{L}$, the algorithm finds a sequence $(\sigma,\theta_{1},...,\theta_{m})$ of formulas such that $\sigma(z_{1},...,z_{m})$ is a formula in the language of Boolean algebras, and each formula $\theta_{i}$ has at most the variables $x_{1},...,x_{n}$ free and where we have the following:

Assume $I$ is an index set, $Z$ is a filter on the set $I$, and $\mathcal{A}_{i}$ is an $\mathcal{L}$-structure for $i\in I$. Assume also that $f_{1},...,f_{n}\in\prod_{i\in I}A_{i}$, and given $1\leq j\leq m$, let $R_{j}=\{i\in I|\mathcal{A}_{i}\models\theta_{j}(f_{1}(i),...,f_{n}(i))\}$. Then we have `\[\prod_{i\in I}\mathcal{A}_{i}/Z\models\phi(f_{1}/Z,...,f_{n}/Z)\]`

if and only if
`\[P(I)/Z\models\sigma(R_{1}/Z,...,R_{m}/Z).\]`

I personally found the above result to be very useful since I was able to determine the sentence algebras and the elementary classes of a certain variety using a version of the above result. Of course, the above result is not as powerful as Łoś's theorem since it does not automatically give you a non-standard model of a structure like the real numbers.