What is difference between these two sentences in terms of meaning :
Tomorrow's game against Arsenal will be a good gauge of their promotion chances
Tomorrow's game against Arsenal will be a good measure of their promotion chances
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An interesting question of overlapping meaning. If we look at the Cambridge dictionary definitions we find that gauge mentions measure six times whereas measure mentions gauge not at all. Therefore, in set terms, measure looks like a subset of gauge. This suggests that gauging something is more than merely measuring it.
I do not tediously repeat all the definitions, which you may easily compare for yourself. What I notice in them is:
Although measure has overtones of non-numerical aspects such as judging importance, quality and effect, the extra meanings of gauge above measure in these entries are to do with judging or estimating, and with showing (in the sense of demonstrating). The extra connotations of gauge are to do with things than cannot be simply measured on a numerical scale: feelings, success, popularity and understanding are all mentioned.
From this perspective, the game will be a good gauge, because it is to do with judging things, such as success, that cannot be numerically or objectively measured.
I hope this helps.