Supporting Lines, Composition and Jensen's Inequality for a Convex Lipschitz Integrand

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space; measurability of real-valued maps on $X$ and integrals are those of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable and Measure Spaces and the Lebesgue Integral: Standing Notation §integral. Write $[0,\infty)$ for the set of nonnegative real numbers, as in Convex Lipschitz Integrands, let $L\in\mathbb{R}$ be nonnegative, and let $\Phi:[0,\infty)\to\mathbb{R}$ be a convex Lipschitz integrand with constant $L$. Then the following hold.

1. (Supporting lines)¶ For every $t_{0}\in[0,\infty)$ there is a real number $s$ with

2. (Composition)¶ Let $h:X\to\mathbb{R}$ be measurable with $0\le h(x)$ for every $x\in X$. Then $\Phi\circ h:X\to\mathbb{R}$ is measurable, and $0\le\Phi(h(x))\le L\,h(x)$ for every $x\in X$. If moreover $h$ is integrable, then $\Phi\circ h$ is integrable and

3. (Jensen's inequality)¶ Suppose that $\mu(X)=1$, and let $h$ be as in claim 2 and integrable. Then $\int_{X}h\,d\mu$ belongs to $[0,\infty)$, and