Suppose that $f$ is a function on $\mathbb{R}\times \mathbb{R^k}$ such that $f(x,\cdot)$ is Borel measurable for each $x\in \mathbb{R}$ and $f(\cdot,y)$ is continuous for each $y\in \mathbb{R^k}$. For $n\in \mathbb{N}$, define $f_n$ as follows.

For $i\in \mathbb{Z}$, let $\displaystyle a_i=\frac{i}{n}$, and for $a_i\leq x \leq a_{i+1}$ let

$\displaystyle f_n(x,y)=\frac{f(a_{i+1},y)(x-a_i)-f(a_i,y)(x-a_{i+1})}{a_{i+1}-a_i}$.

Show that $f_n$ is Borel measurable on $\mathbb{R}\times \mathbb{R^k}$ and $f_n \rightarrow f$ pointwise. Hence show that $f$ is Borel measurable on $\mathbb{R}\times \mathbb{R^k}$. Concluse by induction that every function on $ \mathbb{R^n}$ that is continuous in each variable separately is Borel measurable.

  • $\begingroup$ Can anyone give me a hint. I cannot imagine how to start the solution... $\endgroup$ – user174318 Oct 4 '14 at 22:23

Notice that $$f_n(x,y)=n\sum_{i\in\mathbb Z}\left[f(a_{i+1},y)(x-a_i)-f(a_i,y)(x-a_{i+1})\right]\chi\left\{\left[\frac in,\frac{i+1}n\right)\right\}(x).$$ Define $$f_{n,i}(x,y):=\left[f(a_{i+1},y)(x-a_i)-f(a_i,y)(x-a_{i+1})\right]\chi\left\{\left[\frac in,\frac{i+1}n\right)\right\}(x).$$ This function is Borel measurable on $\mathbb R\times\mathbb R^k$ a sum and product of such functions.

For the pointwise convergence, we have to use the continuity with respect to the first variable and the fact that $$f(x,y)=f(a_i,y)+n(x-a_i)(f(a_{i+1},y)-f(a_i,y)).$$

  • $\begingroup$ How did you derive the expression for $f(x,y)$ like this? $\endgroup$ – user174318 Oct 5 '14 at 23:18
  • $\begingroup$ I translate the condition $a_i\leqslant x\lt a_{i+1}$ in term of characteristic function. $\endgroup$ – Davide Giraudo Oct 7 '14 at 8:29

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