For any $\alpha\in [0, 1]$ and $g:[a, b]\to [0, 1]$, let us define $g^{-}:[0, 1]\to [a, b]$, by the formula $g^{-}(\alpha) = \inf (x\in [a, b] ; g(x)\ge \alpha )$.

Let $f_{n}, f : [a, b]\to [0, 1]$, $n\in \mathbb{N}$, be continuous on $[a, b]$, such that the value $1$ is attained, for $f$ and for all $f_{n}$. It is true that if $f_{n}\to f$ uniformly on $[a, b]$, then $f_{n}^{-}(\alpha)\to f^{-}(\alpha)$, almost everywhere in $[0, 1]$ ?

Thank you in advance.

Best, George