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I need help with the proof of Proposition 7.3.7 ...

Proposition 7.3.7 and its proof read as follows:

In the above proof by Lindstrom we read the following:

" ... ... \(\displaystyle (f + g)^{ -1} ( [ - \infty , r ) ) = \{ x \in X | (f + g) \lt r \}\)

\(\displaystyle = \bigcup_{ q \in \mathbb{Q} } ( \{ x \in X | f(x) \lt q \} \cap \{ x \in X | g \lt r - q \} )\) ... ... " Can someone please demonstrate, formally and rigorously, how/why ...

\(\displaystyle \{ x \in X | (f + g) \lt r \} = \bigcup_{ q \in \mathbb{Q} } ( \{ x \in X | f(x) \lt q \} \cap \{ x \in X | g \lt r - q \} )\) ... ...Help will be much appreciated ...

Peter=============================================================================================================Readers of the above post may be assisted by access to Lindstrom's introduction to measurable functions (especially Lindstrom's definition of a measurable function, Definition 7.3.1) ... so I am providing access to the relevant text ... as follows:

Hope that helps ...

Peter