# On a convex property in the proof of Jensen's inequality

I am reading a proof of Jensen's inequality (the measure theoretic form) in Measure Theory, Probability and Stochastic Processes in Le Gall's book. I know not much about convex functions except the definition, which I believe is not even given in the book, but reads something like (for $$\varphi:\mathbb R\to\mathbb R$$ convex),$$\varphi(tx+(1-t)y)\leq t\varphi(x)+(1-t)\varphi(y),$$for all $$x,y\in\mathbb R$$ and $$t\in[0,1]$$.

Now, here's the theorem.

Theorem 4.3: Let $$(\Omega, \mathcal A,\mu)$$ be a probability space and let $$\varphi:\mathbb R\to\mathbb R_+$$ be a convex function. Then for every $$f\in L^1(\Omega, \mathcal A,\mu)$$, $$\int_\Omega \varphi\circ f\, d\mu\geq\varphi\left(\int_\Omega f\, d\mu\right).$$ Proof: Set $$\mathcal E_\varphi=\{(a,b)\in\mathbb R^2:\forall x\in\mathbb R,\varphi(x)\geq ax+b\}.$$ Then by elementary properties of convex functions, $$\varphi \left(x\right)=\sup_{\left(a{,}b\right)\in \mathcal E_{\varphi }}\left(ax+b\right).\tag1$$ ... ... ...

I do not understand $$(1)$$. Why is this true?

Since $$\phi$$ is convex, for any point $$(z,\phi(z)) \in \mathbb{R}^2$$, there exists an affine function $$f_z(x)=a_zx+b_z$$, such that the line $$L_z$$ which corresponds to $$f_z$$ (i.e. the graph of $$f_z$$) passes through $$(z,\phi(z))$$ and the graph of $$\phi$$ lies above $$L_z$$. In other words $$\phi(x) \geq f_z(x)$$, with equality holding only for $$x=z$$. Let $$A=\{f_z: z\in \mathbb{R}\}$$ be the set of all such affine functions. For any $$x\in \mathbb{R}$$ we have that $$\sup_{f_z\in A}f_z(x) \geq \phi(x),$$ because $$f_x(x)=\phi(x)$$ and $$\sup_{f_z\in A}f_z(x) \leq \phi(x),$$ because each function $$f_z$$ lies below $$\phi$$.
Of course, $$A$$ is in one-to-one correspondence with the set $$\mathcal{E}_{\phi}$$ above and so the claim follows.