Convergence in measure of exp
If the full space is $X$ and your measure is $\mu$, then pointwise $\mu$-a.e. convergence implies convergence in measure if $\mu(X)<\infty$. Here, $\mu = m$, the Lebesgue measure, and $X = [a,b]$, so $m(X) < \infty$. Thus you need only show that you have pointwise a.e. convergence, which you clearly have, since $f_n(x)\to 0$ so long as $x \neq \frac{\pi}{2} + 2\pi n$.