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We use 'holds almost everywhere' because it reflects the essential properties of the function in measure theory, allowing the inequality to fail on a set of measure zero without affecting the overall boundedness classification.

Conditions: The context is measure theory and Lebesgue integration.; The function is measurable.

We use 'holds almost everywhere' because it reflects the essential properties of the function in measure theory, allowing the inequality to fail on a set of measure zero without affecting the overall boundedness classification.

Conditions: The context is measure theory and Lebesgue integration.; The function is measurable.

The essential supremum is defined as the infimum (greatest lower bound) of the set of all essential upper bounds of the function's absolute value.

Conditions: f is a measurable function on E.; M ranges over all real numbers such that |$f(x)$| $\leqslant M$ holds almost everywhere on E.

The essential supremum is defined as the infimum (greatest lower bound) of the set of all essential upper bounds of the function's absolute value.

Conditions: f is a measurable function on E.; M ranges over all real numbers such that |$f(x)$| $\leqslant M$ holds almost everywhere on E.

An ordinary upper bound requires the inequality to hold at every point in the domain, whereas an essential upper bound only requires the inequality to hold almost everywhere, allowing it to fail on a set of measure zero.

Conditions: f is a measurable function on E.; M is a positive real number.

An ordinary upper bound requires the inequality to hold at every point in the domain, whereas an essential upper bound only requires the inequality to hold almost everywhere, allowing it to fail on a set of measure zero.

Conditions: f is a measurable function on E.; M is a positive real number.