Collection of all $T\subseteq\mathbb{R}$ such that $λ^{*}(T) =λ^{∗}(T∩A)$ for all $A\subseteq\mathbb{R}$.

Question

Hi I am doing a past paper and I am stuck on the following question. Let $\theta$ be the collection of all $T\subseteq\mathbb{R}$ such that $λ^{*}(T) =λ^{∗}(T∩A)$ for all $A\subseteq\mathbb{R}$ , show that $\theta$ can be written as $\theta$ = {$T\subseteq\mathbb{R}$ : $λ^{*}(T)\leq m$} and find $m$. The first part of the question asks to give an example of an element of $\theta$ for which I gave the empty set, but I am really unsure of what to do for this part of the question. I am confident that all the elements of $\theta$ are the subsets of $A$ but I don't know how I would use this to answer the question. Any help would be appreciated.

Answer 1

It is given that $∀T∈θ:∀A⊆\mathbb R~(λ^*(T)=λ^*(T∩A))$ in particular $λ^*(T)=λ^*(T∩T^c)$ but $λ^*(T∩T^c)=0$ so $λ^*(T)=0$.

So with that we get that $T∈θ\implies T∈\{B⊆\mathbb R\mid λ^*(B)=0\}$

Now given $T ∈\{B⊆\mathbb R\mid λ^*(A)=0\}$, we know that $T'⊆T$ implies $λ^*(T')=0$, in addition $T∩A⊆T$, so $λ^*(T)=λ^*(T∩A)$ for all $A⊆\mathbb R$, thus $T∈θ$.

So $θ=\{B⊆\mathbb R\mid λ^*(B)=0\}=\{B⊆\mathbb R\mid λ^*(B)≤0\}$