Let $f(x)$ be measurable on $\mathbf{R}^{1}$. If $f(x + 1) = f(x), \mathrm{a.e.} \, x \in \mathbb{R}^{1}$, find the function $g(x)$ on $\mathbf{R}^{1}$ such that $g(x) = f(x), \mathrm{a.e.} \, x \in \mathbb{R}^{1}$, and $g(x) = g(x + 1) \, (x \in \mathbb{R}^{1})$.

Options:
A. Let the point set $E=\{x \in \mathbb{R}^{1} ; f(x) > f(x+1)\}$, and let $\widetilde{E}= \bigcup^{} \left( E+ \langle n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{{f ( x+1 ),}}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ {{{f ( x ),}}} & {{{x \in\widetilde{E},}}} \\ \end{matrix} \right.
$$
B. Let the point set $E=\{x \in \mathbb{R}^{1} ; f(x) = f(x+1)\}$, and let $\widetilde{E}= \bigcap^{} \left( E+ \langle n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{{f ( x ),}}} & {{{x \in\widetilde{E},}}} \\ {{{0,}}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ \end{matrix} \right.
$$
C. Let the point set $E=\{x \in \mathbb{R}^{1} ; f(x) = 0\}$, and let $\widetilde{E}= \bigcup^{} \left( E+ \langle n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{{f ( x ),}}} & {{{x \in\widetilde{E},}}} \\ {{{1,}}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ \end{matrix} \right.
$$
D. Let the point set $E=\{x \in \mathbb{R}^{1} ; f(x) \neq f(x+1)\}$, and let $\widetilde{E}= \bigcap^{} \left( E+ \langle n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{{f ( x ),}}} & {{{x \in\widetilde{E},}}} \\ {{{f ( x+1 ),}}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ \end{matrix} \right.
$$
E. Let the point set $E=\{x \in \mathbb{R}^{1} ; f(x) < f(x+1)\}$, and let $\widetilde{E}= \bigcup^{} \left( E+ \langle n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{{1,}}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ {{{f ( x ),}}} & {{{x \in\widetilde{E},}}} \\ \end{matrix} \right.
$$
F. Let the point set $E=\{x \in \mathbb{R}^{1} ; |f(x)| \neq |f(x+1)|\}$, and let $\widetilde{E}= \bigcup^{} \left( E+ \langle n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{{|f ( x )|,}}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ {{{f ( x ),}}} & {{{x \in\widetilde{E},}}} \\ \end{matrix} \right.
$$
G. Let the point set $E=\{x \in \mathbb{R}^{1} ; f(x) \neq f(x-1)\}$, and let $\widetilde{E}= \bigcup^{} \left( E- \langle n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{{0,}}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ {{{f ( x ),}}} & {{{x \in\widetilde{E},}}} \\ \end{matrix} \right.
$$
H. Let the point set $E=\{x \in \mathbb{R}^{1} ; f(x) \neq f(x+1)\}$, and let $\widetilde{E}= \bigcup^{} \left( E+ \langle n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{{f ( x ),}}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ {{{0,}}} & {{{x \in\widetilde{E},}}} \\ \end{matrix} \right.
$$
I. Let the point set $E=\{x \in \mathbb{R}^{1} ; f(x) \neq -f(x+1)\}$, and let $\widetilde{E}= \bigcup^{} \left( E+ \langle n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{|-f ( x )|,}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ {{{f ( x ),}}} & {{{x \in\widetilde{E},}}} \\ \end{matrix} \right.
$$
J. Let the point set $E=\{x \in \mathbb{R}^{1} ; f(x) \neq f(x+2)\}$, and let $\widetilde{E}= \bigcup^{} \left( E+ \langle 2n \rangle \right)$, then construct the function
$$
g ( x )=\left\{\begin{matrix} {{{f ( x ),}}} & {{{x \overline{{{ \epsilon}}} \widetilde{E},}}} \\ {{{f ( x+1 ),}}} & {{{x \in\widetilde{E},}}} \\ \end{matrix} \right.
$$
Answer: H