Quantum mechanics postulates that an electron is in many places at the same time. Can't we set up experiments to see if it exerts its electromagnetic influence in many places at the same time? (Or its gravitational influence, but that would be much harder to measure.)
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In classical physics the evolution of a measurable quantity, such as the $x$ position of a particle is described by an equation of motion with a solution $x(t)$ such that if you measure the $x$ position of the particle at time $t$ you get the result $x(t)$.
In quantum theory the evolution of a measurable quantity of a single particle such as an electron is described by a matrix. The different possible results of doing a measurement are the eigenvalues of that matrix. If you measure the electron by trying to look for it exerting an electromagnetic force on another particle then you will find it in some particular place when you do that measurement. You will not see it in more than one place.
However, if the electron is left to evolve without exerting an influence on another particle then in general quantum theory sez that what happens to all of the possible measurement results contributes to the result of the experiment: quantum interference. For an example of a measurement of quantum interference with single electrons see
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