I am thinking of the following simple problem:

Suppose that f:RN→Rf:\mathbb{R}^N\rightarrow\mathbb{R} is a continuous, nonnegative function with finite Lebesgue integral. Is it true that

f(y)≤∫RNf(x)dx,∀y∈RN?\begin{equation}

f\left(y\right) \le \int_{\mathbb{R}^N}f\left(x\right)\text{d}x,\quad \forall y\in\mathbb{R}^N\quad\quad?

\end{equation}

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You should not use xx on both sides of the inequality here, since the right hand side does not depend on xx. If the intention is to ask this question with yy instead of xx on the left hand side the answer is no. Try to find a function with the properties you are asking for such that f(1)=1f(1)=1 but ∫f=12\int f = \frac{1}{2}

– Thomas

2 days ago

Obviously… I honestly do not even know why I asked something like that; so obvious…

– underpi

yesterday

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