This paper studies the observability from measurable sets in time for an evolution equation (in a Hilbert space): u' = Au (t >= 0), with an observation operator B. We obtain such an observability inequality in two different settings on (A, B). In the first setting, A generates an analytic semigroup, B is an admissible observation operator for this semigroup, and (A, B) satisfies an observability inequality from time intervals. By the propagation estimate of analytic functions and a telescoping series method (provided in the current paper), we build up the desired inequality for this setting. In the second setting, A generates a Co semigroup, B is a linear and bounded operator, and (A, B) satisfies a spectral-like condition. By methods developed by Phung and Wang [J. Eur. Math. Soc. (JEMS), 15 (2013), pp. 681-703] and Apraiz et al. [J. Eur. Math. Soc. (JEMS), 16 (2014), pp. 2433-2475], we first obtain an interpolation inequality at one time point, and then derive the desired observability inequality for the second setting. This observability inequality is applied to get the bang-bang property of time optimal control problems for several kinds of differential equations.