Consequences of ZF+"all subsets of reals are Lebesgue measurable" (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!
 A: 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.)
A: One can multiply examples, of course. For the amusement of anyone interested, here are three more consequences of there being no nonmeasurable sets (also of the Banach-Tarski decomposition failing).


*

*There are no full finitely-additive conditional probabilities defined for all pairs of subsets of $[0,1]$ (with the second subset non-empty). 

*There are partial orders that don't extend to total orders (and even ones that don't extend to total preorders while keeping strict inequalities).  
These sound negative, like the one about Hahn-Banach failing. Here's a positive-sounding one: 


*If $G$ is a totally ordered abelian group and $\mu$ is a finitely-additive $G$-valued measure (with the obvious definitions, including non-negativity) defined on all subsets of $[0,1]$, then $\mu(A)=0$ for some non-empty $A$. 


My proofs use a modification of the proof that Hahn-Banach implies Banach-Tarski. (By the way, sorry, it's a philosophy and not a math paper that I'm referencing.)
