Can someone solve my non-understandable process in proving a theorem?

Question

Theorem.

Let $E$ be a subset of $\mathbb{R}^n$.

Then, if $p\gt0$, $\int_E|f-f_k|^p\to0$, and $\displaystyle\int_E|f_k|^p\le{}M$ for all $k$, then $\displaystyle\int_E|f|^p\le{}M$.

For your information, $|\cdot|$ means a Lebesgue measure.

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• There is a Theorem A, $$\int_E|f|^p=p\int_0^\infty\alpha^{p-1}w_{|f|}(\alpha)d\alpha$$ ,where $w(\alpha)=\left|\left(\mathbf{x}\in{}E:f(\mathbf{x})\gt\alpha\right\}\right|$
• Since $|f|\le|f-f_k|+|f_k|$ by the triangle inequality, $$\int_E|f|\le\int_E\left(|f-f_k|+|f_k|\right)=\int_E|f-f_k|+\int_E|f_k|$$
• From the Theorem A when $p=1$, $$\int_0^\infty w_{|f|}(\alpha)d\alpha \le \int_0^\infty w_{|f-f_k|}(\alpha)d\alpha + \int_0^\infty w_{|f_k|}(\alpha)d\alpha$$
• Since (?), $$w_{|f|}(\alpha) \le w_{|f-f_k|}(\alpha) + w_{|f_k|}(\alpha)$$
• Thus, \begin{align} \int_E|f|^p &= p\int_0^\infty\alpha^{p-1}w_{|f|}(\alpha)d\alpha \\ &\le p\int_0^\infty\alpha^{p-1}\left(w_{|f-f_k|}(\alpha) + w_{|f_k|}(\alpha)\right)d\alpha \\ &= p\int_0^\infty\alpha^{p-1} w_{|f-f_k|}(\alpha) d\alpha + p\int_0^\infty\alpha^{p-1} w_{|f_k|}(\alpha) d\alpha \\ &=\int_E|f-f_k|^p + \int_E|f_k|^p \end{align}
• Letting $k\to\infty$, $$\int_E|f|^p \le M$$

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I do not understand how the third process causes the fourth process.

Also, I do not understand the last process.

We are proving that $\displaystyle\int_E|f|^p\le{}M$, not proving that $\displaystyle\int_E|f|^p\le{}M$ as $k\to\infty$.

Answer 1

1. Since $p >0$, we have that $$ |f-g|^p \leq |f|^p + |g|^p $$
2. Using 1, we see $$ \int |f|^p dx \leq \int | f- f_k|^p dx + \int |f_k|^p dx $$ for all $k \in \mathbb{N}$.
3. Apply bound, we see $$\int |f|^p dx \leq \int | f- f_k|^p dx + M $$
4. Apply the limit, we see $$\int |f|^p dx \leq\underbrace{ \lim_{k \to \infty} \int | f- f_k|^p dx}_{=0}+ M \leq M $$ as desired.