Computation of an integral in a consequence of the Jensen's inequality.

Question

Proposition. Let $y_1,\dots, y_n\ge 0$, $\alpha_1,\dots, \alpha_n>0$ such that $\sum_{i=1}^n\alpha_i=1$, then $$\prod_{i=1}^ny_i^{\alpha_i}\le \sum_{i=1}^n\alpha_iy_i$$

proof. We consider $\varphi(x)=e^x$, then the Jensen's inequality becomes: $$\exp\bigg(\int_X f\;d\mu\bigg)\le \int_X (\exp f)\:d\mu.$$

Let now $$X=\{t_1,\dots, t_n\},\quad \mathcal{A}=\mathcal{P}(X),\quad \mu(\{t_i\})=\alpha_i\ge 0,$$ when $\sum_{i=1}^n\alpha_i=\mu(X)=1.$ The function $f\colon X\to \overline{\mathbb{R}}$ is define by $$f=\sum_{i=1}^n f(t_i)\chi_{\{{t_i}\}},$$ then $$\int_Xf\;d\mu=\sum_{i=1}^nf(t_i)\mu(\{t_i\}),$$ I have problem to compute the following integral $$\int_X (\exp f)\;d\mu.$$

I know that the result is $$\sum_{i=1}^n\mu(\{t_i\})e^{f(t_i)}$$

Could you give me some hints? Thanks!

Answer 1

It's exactly what you wrote since $e^f = \sum_{i = 1}^{n}e^{f(t_i)}1_{\{t_i\}}.$

Edit: You have a function $f : \{t_1, \dots, t_n\} \to \mathbb{R}$. In general, as you wrote, $f$ has the representation $f = \sum_{i = 1}^{n}f(t_i)1_{\{t_i\}}$. Now apply this result to $e^f : \{t_1, \dots, t_n\} \to \mathbb{R}$ to get $e^f = \sum_{i = 1}^{n}(e^f)(t_i)1_{\{t_i\}}$.

Answer 2

That is not quite the way to do it. First possibility is that you could just $-\log$ the equation to get $$ \sum_{i = 1}^n \alpha_i(-\log(y_i)) \geq -\log\left(\sum_{i = 1}^n \alpha_i y_i \right). $$ But this is true since $g := -\log$ is convex.

But if you really want to use the integral version to prove the above in detail, then set $f: \mathbb{N} \rightarrow \mathbb{R}$ $$ f(i) := y_i, \quad X = \lbrace 1, ..., n \rbrace, \quad \mathcal{A} = 2^X, \quad \mu(\lbrace i \rbrace) = \alpha_i . $$ Then you get $$ \int_X -\log(f)~\mathrm{d}\mu = \sum_{i = 1}^n \alpha_i (-\log(y_i)) $$ Also: $$ -\log\left( \int_X f~\mathrm{d}\mu\right) = -\log\left( \sum_{i = 1}^n \alpha_i y_i \right) $$ We know that $$ -\log\left( \int_X f~\mathrm{d}\mu\right) \leq \int_X -\log(f)~\mathrm{d}\mu. $$