About integrable functions. Let $f_n\colon [0,1]\rightarrow R$ be Lebesgue mensurable with $\int_{0}^{1} |f_n(t)|^3dm(t)<1$ for all $n$. How we can show that $f_n$ is integrable uniformly i.e for all $\epsilon>0$ ther...--prophetes.ai

About integrable functions. Let $f_n\colon [0,1]\rightarrow R$ be Lebesgue mensurable with $\int_{0}^{1} |f_n(t)|^3dm(t)<1$ for all $n$. How we can show that $f_n$ is integrable uniformly i.e for all $\epsilon>0$ there exists $\delta>0$ so that if $E\subseteq[0,1]$ is Lebesgue measurable with $m(E)<\delta$ then $$\int_{E}|f_n(t)|dm(t)<\epsilon$$ for all $n$.

We can use Hölder inequality for $p= 3$ and $q=\frac 32$. Then we have for $n\in\mathbb N$ and $E\subset [0,1]$ measurable $$\int_E |f_n(t)|dm(t)=\int_{[0,1]} |f_n(t)|\mathbf 1_E(t)dm(t)\leq \left(\int_{[0,1]}|f_n(t)|^3dm(t)\right)^{1/3}\mu(E)^{2/3}\leq \mu(E)^{2/3}$$ so for a fixed $\varepsilon$, take $\delta$ such that $\delta^{2/3}\leq \varepsilon$, for example $\delta=\varepsilon^{3/2}$.