| Summary: | In this article, we present a measurable version of the spectral decomposition theorem for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">Z</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>-action on a compact metric space. In the process, we obtain some relationships for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">Z</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>-action with shadowing property and <i>k</i>-type weak extending property. Then, we introduce a definition of measure expanding for a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">Z</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>-action by using some properties of a Borel measure. We also prove one property that occurs whenever a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">Z</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>-action is invariantly measure expanding. All of the supporting results are necessary to prove the spectral decomposition theorem, which is the main result of this paper. More precisely, we prove that if a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">Z</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula>-action is invariantly measure expanding, has shadowing property and has <i>k</i>-type weak extending property, then it has spectral decomposition.
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