TITLE: Shelah's "Can you take Solovay's inaccessible away?"
QUESTION [6 upvotes]: I was wandering if there was a book, thesis or some notes where Shelah's argument for

$\mathtt{ZF}+\mathtt{DC}+$"All sets of reals are Lebesgue measurable" is equiconsistent with $\mathtt{ZFC} + \exists \kappa$ inaccessible
$\mathtt{ZF}+\mathtt{DC}+$"All sets of reals have the Baire property" is equiconsistent with $\mathtt{ZFC}$

contained in Can you take Solovay's inaccessible away? is explained in a newer and/or "more digestible" way. Is there?
Thanks!

REPLY [9 votes]: Chapter 9.5 of the book by Bartoszynski-Judah presents Raisonnier's proof (answering your question 1):
Assume that $\aleph_1$ is not inaccessible in $L$, hence a successor in $L$.  So there is a real $x$ which knows that the $L$-predecessor is countable, hence $\aleph_1^{L[x]}=\aleph_1$, so $X:=\mathbb R \cap L[x]$ is uncountable.
Now assume that  all $\Sigma^1_2(x)$-sets are Lebesgue measurable. Then (this needs some work) the "Raisonier filter" $F_X$ built from $X$, which is a $\bf \Sigma^1_3$-set, is a rapid filter and therefore (again some work, known earlier) not measurable.