Suppose $\{f_n\}$ is a sequence of nonnegative measurable functions on $[0,1]$ such that $f_n(x)\leq K$ for all $n\in \mathbb{N}$ and $x\in [0,1]$. Let $f=\limsup f_n$ and assume that $||f_n||_1\geq ||f||_1$ for each $n$. Then, is it true that $f_n\rightarrow f$ in measure?
It is true via Fatou's Lemma and the uniform bound that $$\int f(x) \;dx \geq \limsup \int f_n(x)\; dx \geq \liminf \int f_n(x)\;dx\geq \int f(x) \; dx,$$ so we get $\lim \int f_n(x)dx=\int f(x)$. This is about where I get stuck. Would it be useful to look at the sets $$A_n=\{x\in [0,1]: f_n(x)>f(x)+\epsilon\} \qquad \text{and} \qquad B_n=\{x\in [0,1]:f_n(x)<f(x)-\epsilon\},$$ since we need to show $\lambda(A_n\cup B_n)\rightarrow 0$ as $n\rightarrow \infty$? Or is there a better way? I would appreciate any hints or solutions.