In Jech's Set Theory, p. 149, one can find the following statement:
''If $B$ denotes the σ-algebra of Borel sets, and if we denote by $C$ the $\sigma$-algebra of sets with the Baire property, and if $I$ is the $\sigma$-ideal of meager sets, we have $B/I = C/I$.''
Why is that? Is it simply because every Borel set has the Baire property?
It’s from the previous two exercises. To show B/I = C/I it is enough to show that any borel set and any set with the Baire property differ by a set in I; ie by a meager set. Clearly if A if Borel it differs from itself (being also Baire) by a meager set (the empty set). Conversely if A is Baire, it differs from a G-delta set (so, borel) it contains by at most a meager set, by the previous exercise.
I’m looking at Old Jech, so pg 506, exercise 39.9