Proof of Lᵖ Inclusions and Constants on a Finite Measure Space

Throughout, $\mathcal{L}^{p}(\mu)$ and $|f|^{p}$ are as in Power-Integrable Functions and the p-Seminorm (ii) and Power-Integrable Functions and the p-Seminorm (i) (so $|f|^{p}\in\mathcal{M}^{+}$ for measurable $f$), with $\mathcal{L}^{1}(\mu)$ the set of integrable functions (Elementary Properties of the p-Seminorm (i)), and $\mu(X)<\infty$ is a real number (Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros (ii)). The constant function $1$ is the indicator $\mathbf{1}_{X}$, and

$$\mathbf{1}_{X}\in\mathcal{M}^{+},\qquad\int_{X}\mathbf{1}_{X}\,d\mu=\mu(X)$$

by Basic Properties of the Integral of Nonnegative Measurable Maps (ii).

Proof of (i). If $p=q$ there is nothing to prove, so let $p<q$ and $f\in\mathcal{L}^{q}(\mu)$; then $f$ is measurable. We claim that

$$|f(x)|^{p}\le1+|f(x)|^{q}\qquad(x\in X).$$

Let $t=|f(x)|$. If $t\le1$, then $t^{p}\le1^{p}=1$ by Properties of Real Powers (vi). If $t>1$, then $t^{q}=t^{p}\,t^{q-p}$ by Properties of Real Powers (iv), and $t^{q-p}>1^{q-p}=1$ by Properties of Real Powers (vi), so $t^{p}<t^{q}$ as $t^{p}>0$ (Properties of Real Powers (ii)). This proves the claim.

By the claim, monotonicity of the integral on $\mathcal{M}^{+}$ (Basic Properties of the Integral of Nonnegative Measurable Maps (iii)) and its additivity (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (i)) give

$$\int|f|^{p}\,d\mu\le\int\bigl(\mathbf{1}_{X}+|f|^{q}\bigr)\,d\mu=\mu(X)+\int|f|^{q}\,d\mu<\infty ,$$

so $f\in\mathcal{L}^{p}(\mu)$.

Proof of (ii). The constant function $c$ is measurable (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii)) and bounded by $|c|$, so it lies in $\mathcal{L}^{p}(\mu)$ by Elementary Properties of the p-Seminorm (viii). For the integral, $\mathbf{1}_{X}$ is a nonnegative measurable real function whose integral as an element of $\mathcal{M}^{+}$ is $\mu(X)<\infty$; by Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi), $\mathbf{1}_{X}\in\mathcal{L}^{1}(\mu)$ and its integral as an element of $\mathcal{L}^{1}(\mu)$ is also $\mu(X)$. The constant function $c$ is the scalar multiple $c\,\mathbf{1}_{X}$, so by linearity of the integral on $\mathcal{L}^{1}(\mu)$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vii))

$$\int_{X}c\,d\mu=c\int_{X}\mathbf{1}_{X}\,d\mu=c\,\mu(X).$$