It has been taught to us that since the black holes have finite mass, their gravitational force is also finite, which totally fits. The Black hole anyhow has a singularity at it's center which is infinitely dense. So if, there's a region inside the black hole having infinite density, how can the total mass of the Black hole be accurately determined?
$\begingroup$ Possible duplicates: physics.stackexchange.com/q/25802/2451 and links therein. $\endgroup$– Qmechanic ♦Sep 16, 2019 at 16:27
The mass of a black hole can be determined in the same way as we determine the mass of any other astronomical object - by observing how it deflects the paths of other objects that pass close to it. These might be objects in closed elliptic orbits (like satellites around planets) or objects in hyperbolic trajectories that are just passing through.
For example, the orbits of several stars orbiting the compact radio source Sagittarius A* at the centre of the Milky Way have been observed over several years, and this gave an estimated mass for Sagittarius A* of 4.31±0.38 million solar masses (see Wikipedia's article on Sagittarius A*). Together with a maximum size estimate for Sagittarius A* (deduced from radio telescope observations), this mass estimate established
beyond doubt that Sagittarius A* must be is probably a supermassive black hole.
We are not sure precisely what lies at the centre of a black hole - the prediction of a singularity is probably an indication that current physics just does not have an adequate model of what happens in such an extreme environment. But whatever the exact structure is, it does not affect what happens outside of the event horizon.
$\begingroup$ @safesphere: the binary black hole LIGO data establishes pretty defintively that at least those objects are black holes. I'm at least unaware of any other way of producing those types of waveforms in the gravitational field. But yes, fair enough point about SgrA* $\endgroup$ Sep 16, 2019 at 16:25
$\begingroup$ @safesphere I didn't think there was any remaining mainstream debate about whether Sagittarius A* is a black hole, but I have edited my answer anyway. $\endgroup$ Sep 16, 2019 at 16:28
$\begingroup$ Well I understand that what you've specified is an indirect method of measuring mass by lensing. But my question isn't how is the mass measured..it's if it is measured, the existence of a singularity is in question? $\endgroup$ Sep 17, 2019 at 7:15
$\begingroup$ @SanikaKhadkikar Your question was "How can the total mass of the Black hole be accurately determined ?" so I answered that. The hypothetical singularity is a point with infinite density, not a region, so the total mass of the black hole is still finite even if it does contain a singularity. $\endgroup$ Sep 17, 2019 at 8:04