Can the voltage across a capacitor be measures with a common volmeter? Why?

Answer 1

It can, just be aware that the voltmeter will draw current and drain away some of the charge. To measure the voltage without drawing current, use a reference voltage and a bridge circuit as a comparator. If the reference voltage exactly matches the capacitor voltage, the meter (usually a galvanometer) will draw no current.

Answer 2

No. A capacitor discharges and charges repeatedly under a load, and to measure one you need the power off. The only problem is that even with the power off the cap still has a charge but it is still discharging and charging. You can only test to see if it is bad (i.e. an open)