From the general postulates by Nielsen and Chuang, 3 talks about measurements.

Quantum measurements are described by a collection ${M_m}$ of measurement operators. These are operators acting on the state space of the system being measured. The index $m$ refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is $\ket{\psi}$ immediately before the measurement then the probability that result $m$ occurs is given by $$p(m) = \Braket{\psi | M_m^\dag M_m | \psi}$$ and the state of the system after the measurement is $$\frac{M_m \ket{\psi}}{\sqrt{\Braket{\psi | M_m^\dag M_m | \psi}}}.$$ The measurement operators satisfy the completeness equation ¹⁾, $$\sum_m{M_m^\dag M_m} = I$$

Special views on measurements have been found to be useful to be adopted.

Projective measurements are observables, they are described by an hermitian operator $M$ with a spectral decomposition of $ M = \sum_m m P_m $. $P_m$ is called the projector on the eigenspace of $M$ with the eigenvalue $m$. $m$ are the possible outcomes of the measurement. The probability for $m$ to occur is then given by $$p(m) = \Braket{\psi | P_m | \psi}$$ and the state of the system after the measurement is $$\frac{P_m \ket{\psi}}{\Braket{\psi | P_m | \psi}}$$.

The POVM is a set of operators ${E_m}$, that are positive and $\sum_m{E_m} = \mathbb{1}$. The probability for a measurement outcome $m$ is then given by $p(m) = \braket{\phi|E_m|\phi}$.