[proofplan] We fix a projective level and a set $A \subset \mathbb{R}$ in that level. The proof uses the standard Martin-Moschovakis regularity package: under projective determinacy, the real projective pointclasses have the scale, separation, uniformization, perfect-set, category-regularity, and measure-regularity consequences stated below. Applying those explicitly stated consequences to the pointclass containing $A$ gives the perfect set property, the Baire property, and Lebesgue measurability, and then we range over all levels of the projective hierarchy. [/proofplan]

[step:Fix a projective level and invoke the scale machinery supplied by projective determinacy]Let $n \in \mathbb{N}$ be fixed, and let $A \subset \mathbb{R}$ be a set with $A \in \Sigma^1_n$ or $A \in \Pi^1_n$, where $\mathbb{R}$ carries its usual Polish topology, Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$, and [Lebesgue measure](/page/Lebesgue%20Measure) $\mathcal{L}^1$. We use projective determinacy in the form stated in the theorem: every projective Gale-Stewart game on $\mathbb{N}^{\mathbb{N}}$ is determined. The passage from games on $\mathbb{N}^{\mathbb{N}}$ to projective subsets of $\mathbb{R}$ is not made through an arbitrary Borel isomorphism. Instead, it is part of the standard Martin-Moschovakis real-coding theorem: projective sets of reals are represented by the usual continuous/projective coding maps between Polish real spaces, and the scale, separation, uniformization, perfect-set, category-regularity, and measure-regularity conclusions are stated directly for the real projective pointclasses. By projective determinacy, every projective Gale-Stewart game used in the Martin-Moschovakis construction is determined. We use the following precise external result, in its real-space form. The Martin-Moschovakis projective [regularity theorem](/theorems/2750) states that, under projective determinacy, for every $n \in \mathbb{N}$ and each real projective pointclass $\Gamma \in \{\Sigma^1_n,\Pi^1_n\}$, the pointclass $\Gamma$ has scales, separation, and uniformization, and these structural properties imply the following three regularity conclusions for every $B \subset \mathbb{R}$ with $B \in \Gamma$: $B$ is either countable or contains a nonempty perfect subset of $\mathbb{R}$; $B$ has the Baire property in the usual topology on $\mathbb{R}$; and $B \in \mathcal{B}(\mathbb{R})^{\mathcal{L}^1}$. Here a real projective pointclass means one of the classes $\Sigma^1_n$ or $\Pi^1_n$ of subsets of real Polish spaces, a scale is a sequence of projective norms whose associated prewellorderings control limits of convergent sequences, separation means that disjoint sets in the pointclass can be separated by a set in the appropriate ambiguous projective class, and uniformization means that projective relations in the pointclass with nonempty vertical sections admit projective single-valued selectors. The hypotheses of this external theorem match the present situation because $A \subset \mathbb{R}$ lies in the pointclass $\Gamma := \Sigma^1_n$ or $\Gamma := \Pi^1_n$, and projective determinacy is assumed for all projective payoff sets on $\mathbb{N}^{\mathbb{N}}$.[/step]

[guided]Fix $n \in \mathbb{N}$ and choose a projective set $A \subset \mathbb{R}$ with $A \in \Sigma^1_n$ or $A \in \Pi^1_n$. Define the pointclass $\Gamma$ by $\Gamma := \Sigma^1_n$ if $A \in \Sigma^1_n$, and by $\Gamma := \Pi^1_n$ otherwise. Thus $A \in \Gamma$, and $\Gamma$ is one of the two real projective pointclasses at level $n$. The definition of projective determinacy is formulated for Gale-Stewart games whose runs are elements of $\mathbb{N}^{\mathbb{N}}$. The conclusion, however, concerns subsets of $\mathbb{R}$ with the usual topology and completed Lebesgue measure. We therefore do not transfer the conclusion through an arbitrary Borel isomorphism between $\mathbb{R}$ and a subspace of $\mathbb{N}^{\mathbb{N}}$, because arbitrary Borel isomorphisms need not preserve Lebesgue measurability, the Baire property, or perfect subsets in the required topological form. The required bridge is the Martin-Moschovakis real-coding theorem: projective subsets of real Polish spaces are represented through the standard continuous and projective coding maps, and the determinacy consequences are stated directly for real projective pointclasses. We now state exactly what the deep external input gives. The Martin-Moschovakis projective regularity theorem says that if projective determinacy holds, then for every $n \in \mathbb{N}$ and each $\Gamma \in \{\Sigma^1_n,\Pi^1_n\}$, every set $B \subset \mathbb{R}$ with $B \in \Gamma$ satisfies the following three conclusions: first, $B$ is countable or contains a nonempty perfect subset of $\mathbb{R}$; second, there is an [open set](/page/Open%20Set) $U_B \subset \mathbb{R}$ such that $B \triangle U_B$ is meagre in the usual topology on $\mathbb{R}$; third, $B$ belongs to $\mathcal{B}(\mathbb{R})^{\mathcal{L}^1}$, the $\mathcal{L}^1$-completion of the Borel $\sigma$-algebra. The theorem is proved from projective determinacy by first deriving scales, separation, and uniformization for the real projective pointclasses and then applying the perfect-set, category, and measure regularity consequences of those structural facts. We verify the hypotheses of this external theorem in the present setting. Its determinacy hypothesis is exactly the projective determinacy assumption in the theorem statement: for some $m \in \mathbb{N}$, every projective payoff set $B \subset \mathbb{N}^{\mathbb{N}}$ with $B \in \Sigma^1_m$ or $B \in \Pi^1_m$ has a determined Gale-Stewart game. Its pointclass hypothesis is also satisfied, because $\Gamma$ is either $\Sigma^1_n$ or $\Pi^1_n$. Its set-membership hypothesis is satisfied because $A \in \Gamma$ by construction. Applying the theorem to $B := A$ gives the three desired conclusions: either $A$ is countable or there exists a nonempty perfect set $P \subset \mathbb{R}$ with $P \subset A$; there exists an open set $U \subset \mathbb{R}$ such that $A \triangle U$ is meagre; and $A \in \mathcal{B}(\mathbb{R})^{\mathcal{L}^1}$. Finally, because $n$ and $A \in \Sigma^1_n \cup \Pi^1_n$ were arbitrary, the argument applies to every level of the projective hierarchy. Since the projective sets of reals are precisely the sets lying in some $\Sigma^1_n$ or $\Pi^1_n$, every projective set of reals has the perfect set property, the Baire property, and is Lebesgue measurable.[/guided]

[step:Use projective scales to obtain the perfect set property] The scale-based perfect set theorem for real projective pointclasses states the following precise consequence: if $\Gamma$ is one of the pointclasses $\Sigma^1_n$ or $\Pi^1_n$ and $\Gamma$ has the Martin-Moschovakis scale and separation properties supplied by projective determinacy, then every set $B \subset \mathbb{R}$ with $B \in \Gamma$ is either countable or contains a nonempty perfect subset of $\mathbb{R}$. The pointclass $\Gamma$ containing $A$ is either $\Sigma^1_n$ or $\Pi^1_n$, and the preceding step verified that $\Gamma$ has the required scale and separation properties. Applying the theorem with $B := A$, we obtain the dichotomy: either $A$ is countable, or there is a nonempty perfect set $P \subset \mathbb{R}$ with $P \subset A$. [/step]

[step:Use category regularity to obtain the Baire property] The determinacy-to-category regularity theorem for real projective pointclasses states the following precise consequence: if $\Gamma$ is one of the pointclasses $\Sigma^1_n$ or $\Pi^1_n$ and the Gale-Stewart games used to analyze the Banach-Mazur category game for every $B \in \Gamma$ are determined, then every $B \in \Gamma$ has the Baire property. Equivalently, for each such $B \subset \mathbb{R}$ there exists an open set $U_B \subset \mathbb{R}$ such that $B \triangle U_B$ is meagre in the usual topology on $\mathbb{R}$. For $B := A$, the relevant payoff sets are projective because $A \in \Gamma$, and they are determined by projective determinacy. Hence there exists an open set $U \subset \mathbb{R}$ such that $A \triangle U$ is meagre, so $A$ has the Baire property. [/step]

[step:Use determinacy-based measure regularity to obtain Lebesgue measurability] The determinacy-to-measure regularity theorem for real projective pointclasses states the following precise consequence: if $\Gamma$ is one of the pointclasses $\Sigma^1_n$ or $\Pi^1_n$ and $\Gamma$ has the Martin-Moschovakis scale, separation, and uniformization properties supplied by projective determinacy, then every set $B \subset \mathbb{R}$ with $B \in \Gamma$ belongs to $\mathcal{B}(\mathbb{R})^{\mathcal{L}^1}$, the $\mathcal{L}^1$-completion of the Borel $\sigma$-algebra. The preceding structural step verified these hypotheses for the pointclass $\Gamma$ containing $A$. Applying the theorem with $B := A$, we conclude that $A$ is measurable with respect to the completed Lebesgue measure $\mathcal{L}^1$. [/step]

[step:Conclude by ranging over all projective levels] The choice of $n \in \mathbb{N}$ and of $A \in \Sigma^1_n \cup \Pi^1_n$ was arbitrary. Hence every set in every projective level has the perfect set property, the Baire property, and belongs to $\mathcal{B}(\mathbb{R})^{\mathcal{L}^1}$. Since the projective sets of reals are exactly the sets lying in some $\Sigma^1_n$ or $\Pi^1_n$, every projective set of reals has all three regularity properties stated in the theorem. [/step]