Soft: Interpretation of a periodic event on circle group Recently I've been exploring probability measures on topological groups, derived from the (essentially) unique Haar measure defined thereon. I had begun to focus on the example of the circle group $S^1$ and i found myself wondering, what would be a "real-world" interpretation of the probability space on $<S^1,\mathfrak{B}_{S^1},\mathfrak{h}_{S^1}>$ (where $\mathfrak{B}_{S^1}$ is the Borel $\sigma$-algebra on $S^1$ and $\mathfrak{h}_{S^1}$ is the Haar measure on $S^1$? It must somehow be periodic in some sense; but how can the in variance of $\mathfrak{h}_{S^1}$ with respect to $S^1$'s action be interpreted in a "real-world" way? * * * An example interpreting this would really help shed some light on this for me; preferably a simple interpretation and noting really exotic.

Note that the Haar measure on $S^1$ is just normalised Lebesgue measure, and this is the quintessential setting of Fourier analysis on compact Abelian groups. A million things have been written on the subject even without a probabilistic interpretation.

The invariance of Haar measure w.r.t. the group action $\theta \rightarrow \theta + \phi$ is simply a statement about the distribution of the measure; namely, it is uniform. Therefore any measurement that can be described by an angle $\theta$ which we expect to be uniform provides the kind of example you seek. For instance, take your pen, throw it up high in the air, and let it drop; it is as likely to point in one direction as any other.