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Both spaces are defined using the same set-builder principle: they are the collections of all measurable functions for which the corresponding norm (either the p-norm or the infinity norm) is finite.

Conditions: The context is the summary of $L^p$ and $L^\infty$ definitions.; Both $\|f\|_p$ and $\|f\|_\infty$ have been defined.

Both spaces are defined using the same set-builder principle: they are the collections of all measurable functions for which the corresponding norm (either the p-norm or the infinity norm) is finite.

Conditions: The context is the summary of $L^p$ and $L^\infty$ definitions.; Both $\|f\|_p$ and $\|f\|_\infty$ have been defined.

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 $0 < p$ < +$\infty$, the p-norm of a measurable function f is defined as the $1/p$ power of the Lebesgue integral of the p-th power of its absolute value over the measurable set E.

Conditions: E is a measurable subset of $\mathbb{R}^n$.; f is a measurable function on E.; $0 < p < +\infty$.

When $0 < p$ < +$\infty$, the p-norm of a measurable function f is defined as the $1/p$ power of the Lebesgue integral of the p-th power of its absolute value over the measurable set E.

Conditions: E is a measurable subset of $\mathbb{R}^n$.; f is a measurable function on E.; $0 < p < +\infty$.