Let (X, F) and (Y, G) be measurable spaces. We need not have measures defined on these spaces in order for the following explanation to hold. A function f : X → Y is called measurable if for every B ∈ G, f-1 B ∈ F. Note that this is a structural notion, and does not depend on the measure.
Again, we see a concept defined in terms of a property of the inverse image of a set.
Consider an example, let X be any space, and F consist only of the empty set and X. Then let f map X to the reals. It is clear that the only such measurable functions are constant on every point in X. For, suppose we consider f-1 ({1}). The singleton set {1} is clearly a measurable set in the Borel space, since it can be written as ∈tersectionn=1 ∈fty [1, 1+1/n). Then f-1 ({1}) has to be either empty, or X. Since f is a function, it follows that there will be exactly one real number r such that f-1 ({r}) will be X. That is, f is the constant function mapping every x ∈ X to r ∈ R.
What does the definition say? It says that the image σ-algebra cannot be "richer" than the pre-image σ-algebra. We can think of a σ-algebra as the collection of "knowable" events. So a function is measurable only if the following holds: B is a knowable event in the image only if f-1 B is knowable in the domain. That is, f cannot create new information - it can only preserve or destroy the information already present in the domain, perhaps by clubbing together domain events into the same range event.