Show that logarithm of Dirichlet function is Lebesgue measurable

Question

I have $\ln (D(x)+1) $. What is the way to prove , that this is a Lebesgue measurable function ?

Answer 1

If f is continuous and g is Lebesgue measurable, $U$ open in $\mathbb R$:

$(f \circ g) ^{-1}(U)=g^{-1}\circ f^{-1} $ Since $U$ is open and $f$ is continuous and $g$ is measurable, we conclude the composition is....

Answer 2

$D$ is measurable for first. Then $D+1$ is measurable because of the composition of the continuous map $u\rightarrow u+1$ with $D$, and $\ln(D(x)+1)$ is measurable because of the composition of the continuous map $v\rightarrow\ln v$ with $D(x)+1$.