Extended validity of weak measurement

We introduce a modified weak value that is related to the mean value of input meter variable. With the help of the modified weak value, the validity conditions for various modified versions of weak value formalism are investigated, in terms of the dependence of the pointer shift on the mean value of the input meter. The weak value formalism, often used to represent the pointer shift, with the modified weak value is of great use in simplifying calculations and giving guidance of practical experiments whenever the mean value of the input meter variable is nonzero. The simulation in a qubit system is presented and coincident well with our theoretical result.