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In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?

Now more carefully, with some notation: Suppose $(X, d_X)$ and $(Y, d_Y)$ are metric spaces, with Borel sets $\mathcal B_X$ and $\mathcal B_Y$, respectively. Let $f:(X, \mathcal B_X) \rightarrow (Y, \mathcal B_Y)$ be a measurable function and let $\mu$ be a probability measure on $(X, \mathcal B_X)$.

Say that $f$ is almost continuous if there is a $\mu$-measure-one Borel set $D \subseteq X$ such that $f$ is continuous on $D$, i.e., the restriction $f|_D:D \rightarrow Y$ is a continuous function where $D$ is given the subspace topology.

As usual, we say that $x \in X$ is a point of continuity (for $f$) if, for every Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ that converges to $x$, we have that $(f(x_n))_{n \in \mathbb{N}}$ is a Cauchy sequence that converges to $f(x)$.

Let $C$ be the set of all points of continuity of $f$. (A classic result [Kechris, I.3.B Prop 3.6] shows that $C$ is a $G_\delta$ set.) Say that $f$ is almost everywhere (a.e.) continuous if $C$ is a $\mu$ measure-one set., i.e. $\mu$-a.e. point is a point of continuity.

Finally, say $g$ is a version of $f$ if $f=g$ $\mu$-a.e.

Clearly, if $f$ is a.e. continuous, then it is almost continuous on the set $C$. The converse does not hold in general: Consider for $f$ the indicator function for the rationals in $[0,1]$. Then $f=0$ on the irrationals, a Lebesgue-measure-one set, but $f$ is discontinuous everywhere.

However, $g=0$ is a version of $f$ and $g$ is a.e. continuous. Which raises the question:

If $f$ is almost continuous, is there a version $g$ of $f$ such that $g$ is a.e. continuous?

Note that by a result of Kuratowski [Kechris, I.3.B Thm. 3.8], we can, possibly changing versions, assume, without loss of generality, that $f$ is continuous on a $\mu$-measure-one $G_\delta$-set $D$. (In the example above, $f$ is almost continuous on the irrationals, which are of course $G_\delta$.)

If it is helpful, one may assume that both $X$ and $Y$ are also complete and separable, i.e., Polish.

[Kechris: "Classical Descriptive Set Theory" 1995]

In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?

Now more carefully, with some notation: Suppose $(X, d_X)$ and $(Y, d_Y)$ are metric spaces, with Borel sets $\mathcal B_X$ and $\mathcal B_Y$, respectively. Let $f:(X, \mathcal B_X) \rightarrow (Y, \mathcal B_Y)$ be a measurable function and let $\mu$ be a probability measure on $(X, \mathcal B_X)$.

Say that $f$ is almost continuous if there is a $\mu$-measure-one Borel set $D \subseteq X$ such that $f$ is continuous on $D$, i.e., the restriction $f|_D:D \rightarrow Y$ is a continuous function where $D$ is given the subspace topology.

As usual, we say that $x \in X$ is a point of continuity (for $f$) if, for every Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ that converges to $x$, we have that $(f(x_n))_{n \in \mathbb{N}}$ is a Cauchy sequence that converges to $f(x)$.

Let $C$ be the set of all points of continuity of $f$. (A classic result [Kechris, I.3.B Prop 3.6] shows that $C$ is a $G_\delta$ set.) Say that $f$ is almost everywhere (a.e.) continuous if $C$ is a $\mu$ measure-one set., i.e. $\mu$-a.e. point is a point of continuity.

Finally, say that a measurable function $g$ is a version of $f$ if $f=g$ $\mu$-a.e.

Clearly, if $f$ is a.e. continuous, then it is almost continuous on the set $C$. The converse does not hold in general: Consider for $f$ the indicator function for the rationals in $[0,1]$. Then $f=0$ on the irrationals, a Lebesgue-measure-one set, but $f$ is discontinuous everywhere.

However, $g=0$ is a version of $f$ and $g$ is a.e. continuous. Which raises the question:

If $f$ is almost continuous, is there a version $g$ of $f$ such that $g$ is a.e. continuous?

If it is helpful, one may assume that both $X$ and $Y$ are also complete and separable, i.e., Polish.

Note that by a result of Kuratowski [Kechris, I.3.B Thm. 3.8], if $Y$ is complete, we can, possibly changing versions, assume, without loss of generality, that $f$ is continuous on a $\mu$-measure-one $G_\delta$-set $D$. (In the example above, $f$ is almost continuous on the irrationals, which are of course $G_\delta$.)

[Kechris: "Classical Descriptive Set Theory" 1995]

In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?

Now more carefully, with some notation: Suppose $(X, d_X)$ and $(Y, d_Y)$ are metric spaces, with Borel sets $\mathcal B_X$ and $\mathcal B_Y$, respectively. Let $f:(X, \mathcal B_X) \rightarrow (Y, \mathcal B_Y)$ be a measurable function and let $\mu$ be a probability measure on $(X, \mathcal B_X)$.

Say that $f$ is almost continuous if there is a $\mu$-measure-one Borel set $D \subseteq X$ such that $f$ is continuous on $D$, i.e., the restriction $f|_D:D \rightarrow Y$ is a continuous function where $D$ is given the subspace topology.

As usual, we say that $x \in X$ is a point of continuity (for $f$) if, for every Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ that converges to $x$, we have that $(f(x_n))_{n \in \mathbb{N}}$ is a Cauchy sequence that converges to $f(x)$.

Let $C$ be the set of all points of continuity of $f$. (A classic result [Kechris, I.3.B Prop 3.6] shows that $C$ is a $G_\delta$ set.) Say that $f$ is almost everywhere (a.e.) continuous if $C$ is a $\mu$ measure-one set., i.e. $\mu$-a.e. point is a point of continuity.

Finally, say $g$ is a version of $f$ if $f=g$ $\mu$-a.e.

Clearly, if $f$ is a.e. continuous, then it is almost continuous on the set $C$. The converse does not hold in general: Consider for $f$ the indicator function for the rationals in $[0,1]$. Then $f=0$ on the irrationals, a Lebesgue-measure-one set, but $f$ is discontinuous everywhere.

However, $g=0$ is a version of $f$ and $g$ is a.e. continuous. Which raises the question:

If $f$ is almost continuous, is there a version $g$ of $f$ such that $g$ is a.e. continuous?

Note that by a result of Kuratowski [Kechris, I.3.B Thm. 3.8], we can, possibly changing versions, assume, without loss of generality, that $f$ is continuous on a $\mu$-measure-one $G_\delta$-set $D$. (In the example above, $f$ is almost continuous on the irrationals, which are of course $G_\delta$.)

[Kechris: "Classical Descriptive Set Theory" 1995]

In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?

Now more carefully, with some notation: Suppose $(X, d_X)$ and $(Y, d_Y)$ are metric spaces, with Borel sets $\mathcal B_X$ and $\mathcal B_Y$, respectively. Let $f:(X, \mathcal B_X) \rightarrow (Y, \mathcal B_Y)$ be a measurable function and let $\mu$ be a probability measure on $(X, \mathcal B_X)$.

Say that $f$ is almost continuous if there is a $\mu$-measure-one Borel set $D \subseteq X$ such that $f$ is continuous on $D$, i.e., the restriction $f|_D:D \rightarrow Y$ is a continuous function where $D$ is given the subspace topology.

As usual, we say that $x \in X$ is a point of continuity (for $f$) if, for every Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ that converges to $x$, we have that $(f(x_n))_{n \in \mathbb{N}}$ is a Cauchy sequence that converges to $f(x)$.

Let $C$ be the set of all points of continuity of $f$. (A classic result [Kechris, I.3.B Prop 3.6] shows that $C$ is a $G_\delta$ set.) Say that $f$ is almost everywhere (a.e.) continuous if $C$ is a $\mu$ measure-one set., i.e. $\mu$-a.e. point is a point of continuity.

Finally, say $g$ is a version of $f$ if $f=g$ $\mu$-a.e.

Clearly, if $f$ is a.e. continuous, then it is almost continuous on the set $C$. The converse does not hold in general: Consider for $f$ the indicator function for the rationals in $[0,1]$. Then $f=0$ on the irrationals, a Lebesgue-measure-one set, but $f$ is discontinuous everywhere.

However, $g=0$ is a version of $f$ and $g$ is a.e. continuous. Which raises the question:

If $f$ is almost continuous, is there a version $g$ of $f$ such that $g$ is a.e. continuous?

Note that by a result of Kuratowski [Kechris, I.3.B Thm. 3.8], we can, possibly changing versions, assume, without loss of generality, that $f$ is continuous on a $\mu$-measure-one $G_\delta$-set $D$. (In the example above, $f$ is almost continuous on the irrationals, which are of course $G_\delta$.)

If it is helpful, one may assume that both $X$ and $Y$ are also complete and separable, i.e., Polish.

[Kechris: "Classical Descriptive Set Theory" 1995]