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The L^$\infty$ space is defined as the set of all measurable functions whose infinity norm is strictly less than positive infinity.

Conditions: f is a measurable function.; The infinity norm $\|f\|_\infty$ has been defined.

The L^$\infty$ space is defined as the set of all measurable functions whose infinity norm is strictly less than positive infinity.

Conditions: f is a measurable function.; The infinity norm $\|f\|_\infty$ has 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.

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.

The $L^p$ space is defined as the set of all measurable functions whose p-norm is strictly less than positive infinity.

Conditions: The p-norm $\|f\|_p$ has been defined.; f is a measurable function.

The $L^p$ space is defined as the set of all measurable functions whose p-norm is strictly less than positive infinity.

Conditions: The p-norm $\|f\|_p$ has been defined.; f is a measurable function.

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.

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.; $0 < p < +\infty$.