Imagine that there was a countably additive and translation invariant measure, , on a Hilbert Space such that every ball has positive measure, finite measure. Because is infinite dimensional, there exists such that . Note that the points and are far apart: , if . Now, the ball, , contains the countable collection of disjoint balls: (you can see where this is going). Thus, because all balls, have the same measure (by translation invariance), and is countable additive, it follows that . Thus, we’ve reached a contradiction.
In some cases, it’s desirable to have an analogue of a measure on a Hilbert space . The most common application is to make statements such as “Property X is true for a.e. function.” One way to do make these statements is through the concept of Prevalence. Another is the definition of a Gaussian like measure called the Abstract Wiener Space.
Note that I wrote this article before realizing that Wikipedia has a similar article.