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

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SUMMARY

The discussion focuses on identifying a function \( f \in L^1(\mathbb{R}) \) such that its maximum function \( Mf \notin L^1(\mathbb{R}) \). The maximum function \( Mf \) is defined as the supremum of \( f \) over a specified domain. The participants seek to understand the conditions under which this scenario occurs, emphasizing the properties of \( L^1 \) spaces and the implications of maximum functions on integrability.

PREREQUISITES
  • Understanding of \( L^1 \) spaces in functional analysis
  • Familiarity with the concept of the maximum function in analysis
  • Knowledge of measure theory and integration
  • Basic principles of supremum and infimum in mathematical analysis
NEXT STEPS
  • Explore the properties of \( L^1 \) spaces and their implications on function behavior
  • Study the maximum function and its role in functional analysis
  • Investigate counterexamples where \( Mf \notin L^1(\mathbb{R}) \)
  • Learn about the relationship between integrability and pointwise limits of functions
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Mathematicians, students of functional analysis, and researchers interested in the properties of integrable functions and maximum functions.

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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