How is the essential supremum defined in terms of essential upper bounds?

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.

Reasoning, step by step

1. Identify all values $M > 0$ such that $|f(x)| \leqslant M$ holds almost everywhere on E.
2. Collect these values M into a set of essential upper bounds.
3. Take the infimum of this set to find the smallest essential upper bound.
4. Define this infimum as the essential supremum of $|f(x)|$ over E.

Example

The video writes the derivation on the board: $\operatorname*{ess\,sup}_{x \in E} |f(x)| = \inf \{ M : |f(x)| \leqslant M, \text{a.e. } x \in E \}$.

Common misconceptions

• Assuming the essential supremum is the maximum value the function actually reaches; it is the infimum of the bounds that hold almost everywhere.
• Confusing the infimum of the set of upper bounds with the minimum; the infimum is the greatest lower bound, which may not be attained by any specific M in the set.

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Explore next

• What is the difference between an essential upper bound and an ordinary upper bound?
• Why do we use 'holds almost everywhere' instead of 'holds at every point'?