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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.

When p = +$\infty$, the infinity norm of a measurable function f is defined as the essential supremum of its absolute value over the set E.

Conditions: $p = +\infty$.; f is a measurable function on E.

When p = +$\infty$, the infinity norm of a measurable function f is defined as the essential supremum of its absolute value over the set E.

Conditions: $p = +\infty$.; f is a measurable function on E.