I’m currently doing some math qualifiers to practice. I came across a problem in the 2000 fall exam at Purdue University that I honestly don’t know how to solve. The problem is the following:
Let $(X, \mathcal{F}, \mu)$ be a finite measure space. Let $\{f_n \}$ be a sequence of measurable functions such that $f_1\in L^1(\mu)$ and $$\mu(\{x\in X:|f_n(x)|>\lambda\}\leq \mu(\{x\in X: |f_1(x)|>\lambda\}),$$ for all $n\in \mathbb{N}=\{1,2,...\}$ and all $\lambda>0$. Prove that $$\lim_{n\to \infty} \frac{1}{n} \int_X \max_{1\leq j\leq n} |f_j(x)|d\mu=0.$$
Any help will be much appreciated! Thank you.
\begin{align*} \frac{1}{n}\int_{X}\max_{1\leq j\leq n}|f_{j}(x)|d\mu&\leq\dfrac{1}{n}\int_{X}\max_{1\leq j\leq n}|f_{j}(x)|1_{|f_{j}|>\lambda}d\mu+\dfrac{1}{n}\int_{X}\max_{1\leq j\leq n}|f_{j}(x)|1_{|f_{j}|\leq\lambda}d\mu\\ &\leq\dfrac{\lambda\mu(X)}{n}+\dfrac{1}{n}\sum_{j=1}^{n}\int_{X}|f_{j}(x)|d\mu_{j},~~~~d\mu_{j}=1_{|f_{j}|>\lambda}d\mu, \end{align*} where \begin{align*} \int_{X}|f_{j}(x)|d\mu_{j}&=\int_{0}^{\infty}\mu_{j}(\{|f_{j}|>\alpha\})d\alpha\\ &\leq\int_{0}^{\infty}\mu_{1}(\{|f_{1}|>\alpha\})d\alpha\\ &=\int_{X}|f_{1}|d\mu_{1}\\ &=\int_{X}|f_{1}|1_{|f_{1}|>\lambda}d\mu \end{align*} as $f_{1}\in L^{1}(\mu)$, the previous term tends to zero as $\lambda\rightarrow\infty$. The rest is clear.