(I'm not sure if this is entirely suitable here so feel free to close it if it's not.) The statement "there is a Lebesgue measure on $\mathbb{R}$($2^\omega$)" means: there is a total $\sigma$-additive monotone (wrt set inclusion) function $\mu$ identical with the usual Lebesgue measure on Cantor space (aka interval [0,1]) so Lebesgue measure is total. It is well known over ZFC Lebesgue measure is not total. 


Some typical consequences of ZFC+ "there is a measure on $\mathbb{R}$" would be negation of CH (as there exists an $\omega\_1$-scale). Of course the measure here is necessarily not Lebesgue.
I'm wondering if there are references about nice consequences of ZF+ "there is a Lebesgue measure on $\mathbb{R}$" in areas other than set theory (or in a more hideous language? Analysis?)? Thanks in advance!



Consequences of ZF+"all subsets of reals are Lebesgue measurable"
There are some useful tables in the back of Gregory Moore's book *Zermelo's Axiom of Choice* showing the deductive relations between several principles that lie between the Axiom of Choice and the existence of a non-measurable set. Some quick examples I see from looking at the tables and taking the constrapositive are that if there are no non-measurable subsets of $\mathbb{R}$, then the Hahn-Banach Theorem fails, not every Boolean algebra has a probability measure, not every infinite set has a non-principal ultrafilter, and not every vector space has a basis.


Are these the kind of consequences you mean?


Also, the result Stadnicki mentioned in the comments is pretty cool, and I haven't seen it before. (the additive groups of $\mathbb{R}$ and $\mathbb{C}$ are no longer isomorphic.)


