I am reading Rudin's RCA and have a question regarding the proof of Jensen's inequality.Let μ be a positive measure on a σ-algbebra M in a set Ω, so that μ ( Ω ) = 1. If f is a real function in L 1 ( μ ), if a &lt; f ( x ) &lt; b for all x ∈ Ω, and if φ is convex on ( a , b ) then φ ( ∫ Ω f d μ ) ≤ ∫ Ω ( φ ∘ f ) d μ .After a few steps Rudin obtains the following inequality: φ ( f ( x ) ) − φ ( t ) − β ( f ( x ) − t ) ≥ 0for every x ∈ Ω. Here, t = ∫ Ω f d μ ∈ ( a , b ) and β is a real number. If φ ∘ f ∈ L 1 ( μ ) then the inequality can be obtained by integrating both sides. However, I am not sure how to proceed in the case φ ∘ f ∉ L 1 ( μ ). Rudin states that in this case, the integral ∫ Ω ( φ ∘ f ) d μ is defined in the extended sense ∫ Ω ( φ ∘ f ) d μ = ∫ Ω ( φ ∘ f ) + d μ − ∫ Ω ( φ ∘ f ) − d μwhere ∫ Ω ( φ ∘ f ) + d μ = ∞ and ∫ Ω ( φ ∘ f ) − d μ is finite.I am unable to show the existence of ∫ Ω ( φ ∘ f ) d μ as defined above.Any help would be greatly appreciated. Thank you.

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I am reading Rudin&#039;s RCA and have a question regarding the proof of Jensen&#039;s inequality.Let   μ be a positive measure on a   σ-algbebra       M   in a set   Ω, so that   μ  (  Ω  )  =  1. If   f is a real function in       L          1        (  μ  ), if   a  &amp;lt;  f  (  x  )  &amp;lt;  b for all   x  ∈  Ω, and if   φ is convex on   (  a  ,  b  ) then  φ            (            ∫          Ω        f    d  μ            )        ≤      ∫          Ω        (  φ  ∘  f  )    d  μ  .After a few steps Rudin obtains the following inequality:  φ  (  f  (  x  )  )  −  φ  (  t  )  −  β  (  f  (  x  )  −  t  )  ≥  0for every   x  ∈  Ω. Here,   t  =      ∫          Ω        f    d  μ  ∈  (  a  ,  b  ) and   β is a real number. If   φ  ∘  f  ∈      L          1        (  μ  ) then the inequality can be obtained by integrating both sides. However, I am not sure how to proceed in the case   φ  ∘  f  ∉      L          1        (  μ  ). Rudin states that in this case, the integral       ∫          Ω        (  φ  ∘  f  )    d  μ is defined in the extended sense      ∫          Ω        (  φ  ∘  f  )    d  μ  =      ∫          Ω        (  φ  ∘  f      )          +          d  μ  −      ∫          Ω        (  φ  ∘  f      )          −          d  μwhere      ∫          Ω        (  φ  ∘  f      )          +          d  μ  =  ∞   and       ∫          Ω        (  φ  ∘  f      )          −          d  μ   is finite.I am unable to show the existence of       ∫          Ω        (  φ  ∘  f  )    d  μ as defined above.Any help would be greatly appreciated. Thank you.

Abigail Palmer · Accepted Answer

Let&amp;nbsp;g&amp;nbsp;(&amp;nbsp;x&amp;nbsp;)&amp;nbsp;=&amp;nbsp;ϕ&amp;nbsp;(&amp;nbsp;t&amp;nbsp;)&amp;nbsp;+&amp;nbsp;β&amp;nbsp;(&amp;nbsp;f&amp;nbsp;(&amp;nbsp;x&amp;nbsp;)&amp;nbsp;−&amp;nbsp;t&amp;nbsp;). Then&amp;nbsp;(&amp;nbsp;ϕ&amp;nbsp;∘&amp;nbsp;f&amp;nbsp;)&amp;nbsp;(&amp;nbsp;x&amp;nbsp;)&amp;nbsp;≥&amp;nbsp;g&amp;nbsp;(&amp;nbsp;x&amp;nbsp;)&amp;nbsp;for all&amp;nbsp;x&amp;nbsp;and&amp;nbsp;g&amp;nbsp;is integrable. Writing&amp;nbsp;ϕ&amp;nbsp;∘&amp;nbsp;f&amp;nbsp;=&amp;nbsp;(&amp;nbsp;ϕ&amp;nbsp;∘&amp;nbsp;f&amp;nbsp;−&amp;nbsp;g&amp;nbsp;)&amp;nbsp;+&amp;nbsp;g&amp;nbsp;we observe that the first term is non-negative and the second term is integrable&amp;nbsp;∫&amp;nbsp;(&amp;nbsp;ϕ&amp;nbsp;∘&amp;nbsp;f&amp;nbsp;)&amp;nbsp;d&amp;nbsp;μ&amp;nbsp;exists.[&amp;nbsp;(&amp;nbsp;ϕ&amp;nbsp;∘&amp;nbsp;f&amp;nbsp;)&amp;nbsp;∧&amp;nbsp;0&amp;nbsp;≥&amp;nbsp;g&amp;nbsp;∧&amp;nbsp;0&amp;nbsp;so&amp;nbsp;(&amp;nbsp;ϕ&amp;nbsp;∘&amp;nbsp;f&amp;nbsp;)&amp;nbsp;−&amp;nbsp;≤&amp;nbsp;g&amp;nbsp;−&amp;nbsp;. Hence&amp;nbsp;∫&amp;nbsp;(&amp;nbsp;ϕ&amp;nbsp;∘&amp;nbsp;f&amp;nbsp;)&amp;nbsp;−&amp;nbsp;d&amp;nbsp;μ&amp;nbsp;≤&amp;nbsp;∫&amp;nbsp;g&amp;nbsp;−&amp;nbsp;d&amp;nbsp;μ&amp;nbsp;&amp;lt;&amp;nbsp;∞&amp;nbsp;]

I am reading Rudin's RCA and have a question regarding the proof of Jensen's inequality. Let &

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