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

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.

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.