[AaronBrown]

The sigma algebra tells you the granularity at which you can distinguish states. In finance, this comes up most often in credit default models. With a standard Merton model of default, defaults are always predictable. You can see the stock price approaching the default barrier, there are no surprise defaults. But suppose you only observed the stock price sampled every second and rounded to the nearest penny. The underlying stock price is still a random walk, but you don't have the full information. In this model, there could be a surprise default.