Question

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 \(a_{i}=i / n\), and for \(a_{i} \leq x \leq a_{i+1}\) let $$ f_{n}(x, y)=\frac{f\left(a_{i+1}, y\right)\left(x-a_{i}\right)-f\left(a_{i}, y\right)\left(x-a_{i+1}\right)}{a_{i+1}-a_{i}} $$ Then \(f_{n}\) is Borel measurable on \(\mathbb{R} \times \mathbb{R}^{k}\) and \(f_{n} \rightarrow f\) pointwise; hence \(f\) is Borel measurable on \(\mathbb{R} \times \mathbb{R}^{k}\). Conclude by induction that every function on \(\mathbb{R}^{n}\) that is continuous in each variable separately is Borel measurable.

Short answer

A function continuous in each variable is Borel measurable.

Step-by-step solution

Construct the Sequence of Functions

We are given the function \(f\) on \(\mathbb{R} \times \mathbb{R}^{k}\) with specific properties. Define \(f_{n}\) for each natural number \(n\) by dividing the real line into intervals \([a_i, a_{i+1}]\) for \(a_i=i/n\). Within each interval, \(f_{n}(x, y)\) is defined as a linear interpolation between \(f(a_i, y)\) and \(f(a_{i+1}, y)\).

Verify Borel Measurability of \(f_n\)

Each \(f_{n}(x, y)\) is a piecewise linear function with respect to \(x\) and is continuous for each fixed \(y\). Since each piece is a continuous function of \(x\) (linear), and because it is constructed as a countable union of these intervals, \(f_{n}\) is Borel measurable in \(\mathbb{R} \times \mathbb{R}^{k}\).

Establish Pointwise Convergence

For each fixed \((x, y)\), \(f_{n}(x, y)\) converges pointwise to \(f(x, y)\) as \(n \to \infty\) because the linear approximation becomes finer, and due to the continuity of \(f(\cdot, y)\). Thus, for any given point, the values of \(f_n(x, y)\) will approach \(f(x, y)\).

Prove \(f\) is Borel Measurable

Since each \(f_n\) is Borel measurable and \(f_{n} \rightarrow f\) pointwise as \(n \to \infty\), the limit \(f\) is Borel measurable. This follows from the fact that pointwise limits of Borel measurable functions are Borel measurable.

Apply Induction to \(\mathbb{R}^n\)

By induction, extend the idea that a function \(f : \mathbb{R}^n \to \mathbb{R}\) that is continuous in each of its variables is, in fact, Borel measurable. Start with the base case \(n = 1\), which is trivially true for continuous functions, and assume it holds for \(\mathbb{R}^{k}\) to show it is true for \(\mathbb{R}^{k+1}\).

Key concepts

Pointwise Convergence

Pointwise convergence is a crucial aspect in understanding the behavior of function sequences. When we say that a sequence of functions \(f_n(x)\) converges pointwise to a function \(f(x)\), we mean that for every fixed point \(x\), as \(n o \infty\), the value \(f_n(x)\) becomes arbitrarily close to \(f(x)\).

In this context, consider the construction of \(f_n\) using linear interpolation on intervals \([a_i, a_{i+1}]\).

• Each \(f_n(x, y)\) is created to approximate \(f(x, y)\) more closely as \(n\) increases.
• This is achieved because the intervals \([a_i, a_{i+1}]\) are shrinking, leading to more precise approximations.

As \(n\) gets larger, \(f_n(x, y)\) converges to \(f(x, y)\) at each point \( (x, y) \), ensuring that the entire sequence approaches \((f(x, y))\) pointwise. This convergence rationale plays a key role in establishing the measurability of \(f\).

Linear Interpolation

Linear interpolation is a numerical method used to estimate the value of a function between two known points. It is based on constructing straight-line segments between these points and using them to estimate values within the interval. In our exercise, linear interpolation helps us define the approximation functions \(f_n\) for each interval \[a_i, a_{i+1}\].

Consider the function structure: within each interval \[a_i, a_{i+1}\], \(f_n(x, y)\) is determined by the formula:

• \[ f_n(x, y)=\frac{f(a_{i+1}, y)(x-a_{i})-f(a_{i}, y)(x-a_{i+1})}{a_{i+1}-a_{i}}\]
• Here, each linear segment is defined by its endpoints \(f(a_i, y)\) and \(f(a_{i+1}, y)\).
• The slope of this line is calculated by the difference in function values divided by the difference in x-coordinates.

Linear interpolation ensures that these interpolated values smoothly connect, providing a continuous approximation of \(f(x, y)\) for each fixed \(y\). This process simplifies the analysis of \(f_n\) and plays a vital role in the broader argument of Borel measurability.

Induction on Functions

Induction on functions is a powerful mathematical technique used to extend conclusions about function properties across different dimensions.
By leveraging the principle of mathematical induction, we can show broader properties for functions in higher dimensions—here illustrating how functions continuous in each variable independently are Borel measurable on \(\mathbb{R}^{n}\).

• Start with a base case, such as \(n = 1\), where it is well-understood that continuous functions are Borel measurable.
• Assume as an induction hypothesis that the property holds true for functions on \(\mathbb{R}^{k}\).
• Prove that if the hypothesis holds for \(\mathbb{R}^{k}\), then it must also hold for \(\mathbb{R}^{k+1}\).

This stepwise proof illustrates that every continuous function in one dimension is Borel measurable, and this property can be transposed across more dimensions by induction. This technique shields the avoidance of starting from scratch in higher dimensional theory and assures comprehensive applicability.