Sigma algebras are an important tool in analysis, but they also get some special attention in probability theory as "sigma fields". Often students studying probability won't have any real background in measure theory, but will know what a sigma field is.
I would hardly qualify as an expert. I can see this as a reasonable separate tag. (Although I suppose that most, in fact almost all, of questions about sigma-algebras would be also tagged measure-theory.)
Thus I think that sigma-fields is a reasonable synonym of sigma-algebras, but I do not thing that sigma algebra should be automatically linked to measure-theory
Do you think it would be useful to ask Willie Wong about his opinion - he created the synonym back than. (I believe at the time he was the moderator who was most active in the tag management.)
Ok, perhaps in $\sigma$-structure mentioned in Halmos or any other typical measure theoretic book. But since you've mentioned $\sigma$-fields, it seems that you can see a few more related structures in the same tag.
@WillieWong In case you happen to be around at some point - what do you think about suggestion to remove the sigma-algebra $\to$ measure-theory synonym? (I thought it might be reasonable to ask you, since you created the synonym.)
@quid I see that in the process you have at least pluralized from sigma-algebra to sigma-algebras. The suggestion to ask about this on meta sounds reasonable.
Although I am not sure how many tag-related posts should be active on meta at the same time - is seems that questions about tags attract little attention on meta as it is.
@B.Mehta Both tags - norm and normed-spaces - are rather large. Personally, I do not think that such substantial change should be made without discussing it on meta first.
The syn is currently at score $+2$. It seems that this one was suggested quite a long time ago, too.
I have no strong feelings. I assume this is brought up for good reason (I haven't been checking Meta.Math.SE most of the time). A lot of the synonyms that made sense 6 or 7 years ago no longer make sense. In general I think that if you and/or other current mods thinks it makes sense to un-do synonym mappings that were put-in when I was a mod, you should feel free to go right ahead!
"A lot of the synonyms that made sense 6 or 7 years ago no longer make sense." I should probably qualify this: I meant that sentence in the way that the site has grown so much that 6/7 years ago it may have made sense to just lump all sigma-algebra questions under measure theory so that they get the "exposure" because having too-specialized tags make things hard to organize and searchable; now the site is much bigger it perhaps pays to have more specialized tags so that the opposite is true.