  This paper studies connections among observable sets, the observability
inequality, the H\"{o}lder-type interpolation inequality and the spectral
inequality for the heat equation in $\mathbb R^n$. We present a characteristic
of observable sets for the heat equation. In more detail, we show that a
measurable set in $\mathbb{R}^n$ satisfies the observability inequality if and
only if it is $\gamma$-thick at scale $L$ for some $\gamma>0$ and $L>0$.We also
build up the equivalence among the above-mentioned three inequalities. More
precisely, we obtain that if a measurable set $E\subset\mathbb{R}^n$ satisfies
one of these inequalities, then it satisfies others. Finally, we get some weak
observability inequalities and weak interpolation inequalities where
observations are made over a ball.
