A simple measure on a set is a function that assigns a non-negative value to subsets of the set, where the value represents the "size" or "measure" of the subset.
The inverse image of a set A under a function f is the set of all elements in the domain of f that map to elements in A. It is denoted as f-1(A).
The inverse image and preimage are often used interchangeably, but they are technically different. The preimage of a set A under a function f is the set of all elements in the domain of f that map to elements in A, while the inverse image is the set of all elements in the domain of f that map to elements in A that also belong to the codomain of f.
Inverse image plays an important role in measure theory as it allows us to define measures on sets that are not necessarily subsets of the domain of the measure. This is useful when dealing with functions that map elements from one set to another, as it allows us to measure the "size" or "measure" of the subsets in the codomain.
Inverse image is used in probability theory to calculate the probability of events. In this context, the inverse image of a set represents the set of all possible outcomes that lead to the event occurring. This allows us to use the measure of the inverse image to calculate the probability of the event.