MHB Solving for $Mf \notin L^1(\mathbb{R})$ with $f \in L^1 (\mathbb{R})$

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To find a function f in L^1(ℝ) such that Mf, the maximum function, is not in L^1(ℝ), one must understand the properties of the maximum function and its relationship with integrable functions. The maximum function Mf is defined as the pointwise maximum of the function f over a certain domain. A common example involves constructing f to be integrable but with unbounded or oscillating behavior that prevents Mf from being integrable. The discussion highlights the need for examples and deeper analysis of the conditions under which Mf fails to be in L^1(ℝ). Understanding these concepts is crucial for solving the posed problem.
mathmari
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Hello! :o

How can we find a $f \in L^1 (\mathbb{R})$ such taht $Mf \notin L^1 ( \mathbb{R})$, where $Mf$ is the maximum function ?? (Wondering)
 
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