10. Let \( f \) be a bounded measurable function on a set of finite measure \( E \). For a measurable subset \( A \) of \( E \), show that \( \int_{A} f=\int_{E} f \cdot \chi_{A} \).
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We need to show that the integral of a function \( f \) over a subset \( A \) of a set \( E \) is equal to the integral of \( f \) multiplied by the characteristic function of \( A \) over the entire set \( E \). Show more…
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