Conditional expectation proof.

Question

I am reviewing in a book the proof of the following equation:

E(1_A |{∅,B^C,B,Ω})=P(A│B)1_B+P(A│B^C )1_B^C

The proof begins by saying "Note that every measurable function with respect to the σ-algebra G (for E (X | G)) given by {∅,B^C,B,Ω} is constant in both B^C and B.

My question is: Why every measurable function with respect to the σ-algebra G is constant in both B^C and B?

Answer 1

Let be measurable w.r.t $G$. If $f$ takes two distinct values $a,b$ on $B$ then $B\cap f^{-1} \{a\}$ and $B\cap f^{-1} \{b\}$ must both be in the given sigma algebra. These sets are non-empty, disjoint and they are both contained in $B$. There are no such subsets of $B$ in the sigma algebra so $f$ cannot attain more than one value on $B$. Similarly for $B^{c}$.