# There is no infinite dimensional Lebesgue measure

Imagine that there was a countably additive and translation invariant measure, ${m}$, on a Hilbert Space ${\mathcal{H}}$ such that every ball ${B(v, \varepsilon)}$ has positive measure, finite measure. Because ${\mathcal{H}}$ is infinite dimensional, there exists ${\{e_{i} | i = 1, \cdots, \infty\}}$ such that ${\langle e_{i}, e_{j}\rangle = \delta_{ij}}$. Note that the points ${e_{i}}$ and ${e_{j}}$ are far apart: ${\|e_{i} - e_{j}\|^2 = \|e_{i}\|^2 + \|e_{j}\|^2 = 2}$, if ${i \neq j}$. Now, the ball, ${B(0, 2)}$, contains the countable collection of disjoint balls: ${\{B(e_{i}, \frac{1}{2}) | i = 1,\cdots, \infty\}}$ (you can see where this is going). Thus, because all balls, ${B(e_{i}, \frac{1}{2})}$ have the same measure (by translation invariance), and ${m}$ is countable additive, it follows that ${\infty = \sum_{i=1}^\infty m(B(e_{i}, \frac{1}{2})) < B(0, 2) < \infty}$. Thus, we’ve reached a contradiction.

In some cases, it’s desirable to have an analogue of a measure on a Hilbert space ${\mathcal{H}}$. 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.