an application of Martin's axiom to Lebesgue measure I am an beginner in set theory and try to solve an exercise in the 2nd chapter in Kunen's book:
Assume $MA(\kappa)$, let $A$ be a family of Lebesgue measurable subsets of $\mathbb{R}$,with $|A|=\kappa$. Show that $\cup{A}$ is Lebesgue measurable and $\mu(\cup{A})=\mu(\cup{B})$ for some countable $B\subset A$.
Could you help me with this. Thank you!
By the way, maybe another soft question: I can solve the problem if the forcing notion has been provided  in the hint of the problems and I generally do not know how to construct a specific partial order to solve corresponding questions. Is there some way to improve this ability? 
 A: This readily follows from the fact that MA($\kappa$) implies that the union of fewer than $\kappa$ Lebesgue null sets is null: Suppose $\theta < \kappa$ and $\{A_i : i < \theta\}$ are Lebesgue measurable. We argue by induction on $\theta$. If $\theta \leq \omega$ this is clear so assume that it is uncountable. Put $X_i = \bigcup \{A_j : j < i\}$. Then each $X_i$ is Lebesgue measurable by inductive hypothesis. If cofinality of $\theta$ is $\omega$, we easily finish. If not, then since $\mu(X_i)$'s are monotonically increasing reals, they stop growing at some point (why?) which means that $X_{i+1} \setminus X_i$'s are eventually null and the result follows. Finally, note that if countably many $A_i$'s cannot cover $\bigcup \{A_i : i < \theta\}$ modulo null, then we can inductively build a strictly increasing sequence of real numbers of order type $\omega_1$.
On your soft question: The forcing for proving the additivity of null ideal is the amoeba forcing. It is natural to an analyst. When you are applying forcing to construct a model satisfying a statement from a branch of math, it is natural that some knowledge of that branch will play a role. For example, for amoeba forcing, the proof of cccness uses the fact that the space $L^1[0, 1]$ is separable. This is quite an easy example; a more interesting one is the forcing used to get a model where every strong measure zero set is countable.
Most set-theorists, especially when they are young, work on a small class of problems and hence specialize into forcings that are relevant to their questions. Others, like Shelah, work on enormously diverse questions.
