Proving function is well-defined over set of equivalence classes

Question

This is a subproblem of a problem I am trying to solve.

For a measure space $(X,\mathcal{A}, \mu)$ define $\backsim$ as the relation on $\mathcal{A}$ where $A \backsim B$ iff $\mu(A \triangle B)=0$

First I need to prove that this is an equivalence relation. I already did this.

Then, define $\mathcal{B}$ to be the set of equivalence classes for this relation. Prove that $\mu$ is well-defined on $\mathcal{B}$.

I am always confused by proofs of well-definedness. Do I start with two elements of the same equivalence class and then two more elements from a different equivalence class? What would need to hold to prove that $\mu$ is well defined?

Answer 1

The statement '$\mu$ is well defined' means that if $A \sim B$ then $\mu (A)=\mu(B)$. On other words $\mu (A\Delta B)=0$ implies $\mu (A)=\mu(B)$. To prove this note that $\mu (A) =\mu (A \cap B) +\mu (A\setminus B) \leq \mu (B)+ \mu (A\Delta B) =\mu (B)$. Simialrly, $\mu (B) \leq \mu (A)$.

Answer 2

Let $X$ be a set and let $\mathcal P$ be a partition of set $X$.

Then a function $f:X\to Y$ might induce a function $\hat f:\mathcal P\to Y$ that is prescribed by $P\mapsto f(x)$ where $x$ denotes a representative of $P$ (i.e. $x\in P$).

This however will only be the case if every element $y$ of $P$ will give the same value $f(y)$.

(If not then we get $P\mapsto y$ and $P\mapsto x$ with $x\neq y$)

Checking whether that is indeed the case is checking whether $\hat f$ is well-defined.

So actually it comes to checking that $f$ is constant on every $P\in\mathcal S$.