Measurable functions : Is X F Measurable?

Question

Example 1: Let F denote the σ-field on Ω = {1, 2, 3, 4, 5, 6} generated by {1, 2}, {1, 4} and {2, 3, 5} and let X = 2 · I{1,2} + 3 · I{1,3,5} − 2 · I{3,5} . Is X F measurable?

Try : I got X(1) = 5 , X(2) = 2 , X(3)=X(5)=1 , X(4)=0

So I got X = 5.I{1} + 2.I{2} + 1.I{3,5}

Note : I is the indicator function and the curly brackets next to I are in subscripts.

Answer 1

$\{1,2\}$ is an element of $\mathcal{F}$, as is $\{3,5\}=\{2,3,5\}\setminus \{1,2\}$ and $\{1,3,5\}=\{3,5\}\cup(\{1,2\}\cap \{1,4\})$.

Therefore each indicator function in the definition of $X$ is $\mathcal{F}$-measurable, hence $X$ is $\mathcal{F}$-measurable.