The Maximum and Minimum Functions as Upper Functions

The Maximum and Minimum Functions as Upper Functions

Recall from The Maximum and Minimum Functions of Two Functions page that if $f$ and $g$ are functioned defined on $I$ then the maximum function of $f$ and $g$ denoted $\max (f, g)$ is defined for all $x \in I$ by:

(1)
\begin{align} \quad \max (f, g)(x) = \max \{ f(x), g(x) \} \end{align}

Similarly, the minimum function of $f$ and $g$ denoted $\min (f, g)$ is defined for all $x \in I$ by:

(2)
\begin{align} \quad \min (f, g)(x) = \min \{ f(x), g(x) \} \end{align}

We will now see that if $f$ and $g$ are also upper functions on $I$ then the maximum and minimum functions of $f$ and $g$ are also upper functions on $I$.

 Theorem 1: Let $f$ and $g$ be upper functions defined on $I$. Then $\max (f, g)$ and $\min (f, g)$ are both upper functions on $I$.
• Proof: Let $f$ and $g$ be upper functions defined on $I$. Then there exists increasing sequences of step functions $(f_n(x))_{n=1}{\infty}$ and $(g_n(x))_{n=1}^{\infty}$ that converge to $f$ and $g$ (respectively) almost everywhere and such that $\displaystyle{\lim_{n \to \infty} \int_I f_n(x) \: dx}$ and $\displaystyle{\lim_{n \to \infty} \int_I g_n(x) \: dx}$ are finite.
• Consider the sequence $(\max (f_n, g_n))_{n=1}^{\infty}$. This is an increasing sequence of step functions since $(f_n(x))_{n=1}^{\infty}$ and $(g_n(x))_{n=1}^{\infty}$ are both increasing sequences of step functions. Furthermore, $(\max (f_n, g_n))_{n=1}^{\infty}$ converges to $\max (f, g)$ almost everywhere on $I$.
• Consider the limit $\displaystyle{\lim_{n \to \infty} \int_I \max (f_n(x), g_n(x)) \: dx}$. We need to show that this limit is finite. Recall that $\max (f, g) + \min (f, g) = f + g$. So:
(3)
\begin{align} \quad \lim_{n \to \infty} \int_I \max (f_n, g_n) \: dx = \lim_{n \to \infty} \int_I [f_n + g_n - \min (f_n, g_n)] \: dx = \int_I f_n(x) \: dx + \int_I g_n(x) \: dx - \lim_{n \to \infty} \int_I \min (f_n, g_n) \: dx \end{align}
• However, we see that since $(f_n(x))_{n=1}^{\infty}$ and $(g_n(x))_{n=1}^{\infty}$ are both increasing sequences of functions that converge to $f$ and $g$ respectively, that then $\min (f_n, g_n) \leq f$ for all $n \in \mathbb{N}$. So $\displaystyle{\int_I \min (f_n, g_n) : dx}$ is finite, so $\min (f, g)$ is an upper function, and from the equality above we see that the righthand side is finite, so $\max (f, g)$ is also an upper function. $\blacksquare$