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Here is another solution.

Define $A_j=\{x:|f(x)|\geq j \},\forall j\in \mathbb N$ .

The given statement is clearly true if $|f|$ is bounded. So if we can show the integrals over the defined sets tend to zero, we are done.

Now, $ \chi_{_{A_j}} $ is a collection of monotonically decreasing sequence with limit $0$. And so is true for $ |f| \chi _{_{A_j}} $ with $|f|\chi _{_{A_1}}\in L^1(\mu)$. So apply exercise 1.7 of Rudin or just apply DCT to get $\lim_{j \to \infty}\int_{A_j}|f| d\mu=0 $.

Here is another solution.

Define $A_j=\{x:|f(x)|\geq j \},\forall j\in \mathbb N$ .

The given statement is clearly true if $|f|$ is bounded. So if we can show the integrals over the defined sets tend to zero, we are done.

Now, $ \chi_{_{A_j}} $ is a collection of monotonically decreasing sequence with limit $0$. And so is true for $ |f| \chi _{_{A_j}} $ with $|f|\chi _{_{A_1}}\in L^1(\mu)$. So apply Dominated Convergence Theorem to get $\lim_{j \to \infty}\int_{A_j}|f| d\mu=0 $.

Here is another solution.

Define $A_j=\{x:|f(x)|\geq j \},\forall j\in \mathbb N$ .

The given statement is clearly true if $|f|$ is bounded. So if we can show the integrals over the defined sets tend to zero, we are done.

Now, $ \chi_{A_j} $ is a collection of monotonically decreasing sequence with limit $0$. And so is true for $ |f| \chi _{A_j} $ with $|f|\chi_{A_1}\in L^1(\mu)$. So apply exercise 1.7 of Rudin or just apply DCT to get $\lim_{j\rightarrow \infty}\int_{A_j}|f| d\mu=0 $.

Here is another solution.

Define $A_j=\{x:|f(x)|\geq j \},\forall j\in \mathbb N$ .

The given statement is clearly true if $|f|$ is bounded. So if we can show the integrals over the defined sets tend to zero, we are done.

Now, $ \chi_{_{A_j}} $ is a collection of monotonically decreasing sequence with limit $0$. And so is true for $ |f| \chi _{_{A_j}} $ with $|f|\chi _{_{A_1}}\in L^1(\mu)$. So apply exercise 1.7 of Rudin or just apply DCT to get $\lim_{j \to \infty}\int_{A_j}|f| d\mu=0 $.

Here is another solution.

Define $E_j=\{x:|f(x)|\geq j \},\forall j\in \mathbb N$ .

The given statement is clearly true if $|f|$ is bounded. So if we can show the integrals over the defined sets tend to zero, we are done.

Now, $ \chi_{E_j} $ is a collection of monotonically decreasing sequence with limit $0$. And so is true for $ |f| \chi _{A_j} $ with $|f|\chi_{A_1}\in L^1(\mu)$. So apply exercise 1.7 of Rudin or just apply DCT to get $\lim_{j\rightarrow0}\int_{A_j}|f| d\mu=0 $.

Here is another solution.

Define $A_j=\{x:|f(x)|\geq j \},\forall j\in \mathbb N$ .

The given statement is clearly true if $|f|$ is bounded. So if we can show the integrals over the defined sets tend to zero, we are done.

Now, $ \chi_{A_j} $ is a collection of monotonically decreasing sequence with limit $0$. And so is true for $ |f| \chi _{A_j} $ with $|f|\chi_{A_1}\in L^1(\mu)$. So apply exercise 1.7 of Rudin or just apply DCT to get $\lim_{j\rightarrow \infty}\int_{A_j}|f| d\mu=0 $.