This talk introduces the observability inequality for heat equations defined on a bounded domain of $\R^d$ and the whole space $\R^d$ respectively, where the observation sets are measured by a Hausdorff content, defined by a log-type gauge function, which is closely related to the heat kernel. For the heat equation on a bounded domain, we obtain the observability inequality for observation sets of positive log-type Hausdorff content, which, in particular, implies the observability inequality for observation sets of positive s-dim Hausdorff measure, where s can be any number in $(d-1,d]$. For the heat equation over $\R^d$, we build up the observability inequality for observation sets which are thick at scale of the log-type Hausdorff content. This is a recent work joint with H. Shanlin and M. Wang.