It is inadmissible to prove mathematical concepts using real world examples. They are not mathematics. The power of math is in that it is abstracted from many examples in real world, and, hence is in position to have and use its own systems of axioms.
Measure theory, and union sets as an example, can not tell you why you make union of sets, but it can tell you, once union is made, what properties in relations to other sets these sets have.
If math was about real world examples and if math depends on them, real world things will be used to prove mathematical theorem. But they are not! Theorems are proved using ONLY mathematical concepts, axioms, and other, already proved theorems. So, what are we getting in math from real world, in relation to axiomatic, postulate, theorems structure within mathematics?
In most cases we are getting ready to use counts, numbers from real world, then specified and requested sequences of numerical operations to be performed on numbers or sets of numbers. These are starting points for further mathematical calculations. In mathematics, these counts, numbers, numerical relationships, coming from real world, are either provable directly from ZFC axioms or are postulates that serve as starting points anyway (provable by other theorems). These are starting points for mathematical development in certain directions (like polynomials, differential equations, fractions) and require no proof within math, or at least, as I indicated, they directly follow from ZFC axioms. In other words, they are "given", they are "assumed", "supposed" etc. Of course, a huge amount of logic is activated and is used just to make these starting points to mathematics available, and this logic is in the World # 1, world outside mathematics, like physics, economics, engineering, probability experiments, trading, finance, chemistry, etc.
Measure theory is interested in how to measure size of a set. Note that it does not care where from real world these sets come from. They can come from hundreds of different sources, from hundreds of things, objects we can count. But, once we find out that certain set is common to all those sources, we can focus to investigate the properties of a set as a separate concept. Hence, property of a set can not be a flavour of apples counted, or make a car from cars counted. Properties of a set can not be the color of shirts we have just counted. The only property of set is how many elements it has. All other information about sets (for instance what we have counted, and why) is outside mathematics, but it is important for us to keep track what we have counted or measured, so we can apply results of sets manipulation back to our practical application.
Measure theory deals with comparison of sets. And there is a lot to do there. You can have union of sets, intersection of sets, set complement. You can pair elements of sets and get different sizes, i.e. cardinality. You compare cardinality of sets by matching elements of one set with elements of the other. Don't forget, the main property, during any set manipulation, is cardinality of set. Of course, measure theory talks a lot about subsets, and collection of subsets of a set. But note, it is still manipulation of set concept. No real world concepts will enter in any part of measure theory.
Sets. Subsets. Lebesgue Measure. Lebesgue Integral. Sigma Algebra. Algebra of Sets.
[ applied math, applied mathematics ]