What is the difference between an essential upper bound and an ordinary upper bound?

Conditions

• f is a measurable function on E.
• M is a positive real number.

Reasoning, step by step

1. Define an ordinary upper bound: $|f(x)| \leqslant M$ for all $x \in E$.
2. Define an essential upper bound: $|f(x)| \leqslant M$ for almost every $x \in E$.
3. Contrast the two by noting that the essential upper bound permits exceptions on a set of measure zero.
4. Conclude that 'essential' refers to the property holding almost everywhere rather than everywhere.

Example

The video explicitly contrasts them: 'If it holds for every x, then M is an upper bound... However, what does essential upper bound mean here? We do not require it to hold at every point—everywhere—but only almost everywhere.'

Common misconceptions

• Interpreting 'essential upper bound' as requiring the inequality to hold at every point of E.
• Assuming that a function can only be bounded if it is bounded everywhere; the video introduces 'essentially bounded' to handle functions that may spike on measure-zero sets.

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