# Lebesgue Integral Computation Involving Infimum

Problem. Suppose $$f$$ is nonnegative, $$\int_{\mathbb{R}^n} f = 1$$, and $$f(x) \leq 1$$ for all $$x \in \mathbb{R}^n$$. Calculate the following Lebesgue integral

$$\inf \int_{\mathbb{R}^n}\|x\|_2^2f(x)dx,$$

with the infimum taken over all functions $$f$$.

My idea was to define $$f_k(x) := x_k^2f(x)$$ such that

$$\int_{\mathbb{R}^n}\|x\|_2^2f(x)dx = \int_{\mathbb{R}^n}(x_1^2 + \dots + x_n^2)f(x) dx= \int_{\mathbb{R}^n} \sum_{k=1}^n f_k.$$

But I don't see a way to use MCT or DCT since the series is finite, so I'm not sure if this is the right approach?

$$\textbf{Theorem:}$$ If, $$(\Omega,\mu)$$ is a sigma-finite measure space and $$G:\Omega \to \mathbb{R}$$ is a measurable functions with sublevels of finite measure (i.e., $$\mu(\{G < t\}) < \infty$$ for all $$t \in \mathbb{R}$$) then the infimum of the functional $$J(f) := \int_{\Omega} Gf\,d\mu$$ in the class of functions $$\displaystyle \mathscr{C} = \left\{f: 0 \le f \le 1, \int_{\Omega} f\,d\mu = m\right\}$$ is attained by $$f = \chi_{\{G < s\}} + c\chi_{\{G = s\}}$$ where, $$s = \sup \{t: \mu(\{G < t\}) \le m\}$$ and $$c\mu(\{G = s\}) = m - \mu(\{G < s\})$$.
(The proof is left as an exercise and I can help you along the way if you need it. The strict monotonicity of $$G(x) = |x|^2$$ should make the proof much simpler in this case.)