Measure of the set $\{t: \sqrt2-t \in \mathbb Q^c\}$

Let $f$ be the characteristic function of irrationals in $[0,1]$. I want to compute estimate the integral $$\int_{\sqrt2-1}^{-1/2}f(\sqrt2-t)\ dt.$$

I reduced the integral to the measure of the set $[\sqrt2-1,-1/2]\cap\mathbb Q^c\cap \{t: \sqrt2-t \in \mathbb Q^c\}$.

Using this can I say for sure that the integral must be strictly less than $1/2-\sqrt 2$ or is equality possible?

I'm stuck in saying anything about measure of the last set above.