Measurability of maximal functions

Question

I have been reading Fourier Analysis by J. Duoandikoetxea. I got stuck on proving the measurability of maximal functions. The general maximal function/operator in this book is from the following proposition:

The theorem as it is stated is, in my opinion, simply false; I don't see how $T^*f$ is measurable. (PS. The definition is that $T_t$ and $T^*$ should map $L^p$ functions to measurable functions $X\to\mathbb{C}$.) Incidentally: Well, the proof of this particular theorem carries over verbatim if we use Lebesgue outer measure instead. However, later in the book the Marcinkiewicz interpolation theorem is used for these maximal functions, so it seems that we have to prove its measurability...

For specific functions (such as the famous Hardy–Littlewood maximal function), the measurability can be proved directly (in this case, one observes that it suffices to take rational $t$). In general, I need the measurability of the following type of maximal functions (as is used later in the book):

Let $\phi\in L^1(\mathbb{R}^n)$ such that $\int\phi=1$. Let $\phi_t=t^{-n}\phi(x/t)$. Then $T^*f(x):=\sup_{t\in\mathbb{R}}|\phi_t*f(x)|$ is measurable.

Is this true?

Answer 1

On the special case you mention at the end: If you know $f\in L^\infty,$ then $\phi_t*f$ is continuous, hence so is $|\phi_t*f|.$ We are then taking the supremum of a familiy of continuous functions, which is lower-semicontinuous, hence measurable.

Answer 2

The problem is not even so much measurability, as well-definedness. For example let $$T_t\colon L^p(\mathbb{R})\to L^p(\mathbb{R}) ,\,T_t f(x)=f(x-t).$$ If $f=0$, then $\sup_{t\in\mathbb{R}}T_t f(x)=0$ for all $x\in\mathbb{R}$, but if $f=1_{\{0\}}$, then $\sup_{t\in\mathbb{R}}T_t f(x)=1$ for all $x\in \mathbb{R}$. So functions in the same equivalence class don't give rise to maximal functions in the same equivalence class. This is why you shouldn't define suprema of uncountable families pointwise.

Instead, the correct definition is as follows. Let $L^+(X,\mu)$ be the equivalence classes of measurable functions $X\to[0,\infty]$ and define a partial order on $L^+(X,\mu)$ by $[f]\leq[g]$ if $f\leq g$ a.e. This clearly is independent of the representatives. Under suitable conditions on the measure space (for example if $(X,\mu)$ is $\sigma$-finite), every non-empty subset of $L^+(X,\mu)$ has a least upper bound with respect to $\leq$. The correct definition of the supremum of an uncountable family is then the supremum in this partially ordered space, which is by definition an equivalence class of measurable functions.

These kind of questions are treated in Fremlin's books on measure theory. Chapters 21 and 36 should be the most relevant one for this particular question.