The basic fact of expansion is fairly easily measured.
We have a great deal of confidence how most stars like our sun works. The emission of light from these stars have a “frequency pattern” in them which indicates things like hydrogen. So this pattern is purely dependent on the element (say hydrogen) and is “fixed”.
By observing the light from stars from far away, we can also detect this pattern meaning it confirms that it is almost certainly hydrogen. However this pattern is “shifted” by a certain amount. This shift tells us that the source of this light is either moving away or towards us and also gives an estimate of how fast they are moving away or towards us.
It happens that a general pattern emerges when looking at light from many different galaxies in many different directions. The further away we believe the galaxy to be, the faster it appears to be moving away from us (in general). This most consistent and simple explanation of this is an expanding universe.
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