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I'm asked in an exercise to prove every integrable real function $f : R \rightarrow R$ on an interval $[a,b]$ has a graph that's a null set. I would appreciate some assistance with this!

More information: Integrable here means Darboux/Riemann integrable. This is not a measure theory question and I am unfamiliar with measure theory results and theorems.

We have already proven that if f is continuous, then its graph will be a null set. I was trying to generalize the ideas from that proof to this proof but I haven't succeeded yet... I really have no idea how to approach this.

Thanks a lot!

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    @daro: You don't need to decompose or think about negative and positive values. The difference between the largest value of the function and the smallest value of the function is always positive, and represents the height of a certain rectangle.2012-05-23

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(Note: I'm assuming below that $f$ is bounded; Riemann integrability is usually only considered for bounded functions, so this is not an unreasonable assumption.)

Given a function $f$, an interval $[a,b]$, and a partition $P$ of $[a,b]$, let us denote by $\overline{S}(f,P)$ the upper Riemann sum of $f$ on $[a,b]$ relative to the partition $P$; that is, we consider $\overline{S}(f,P) = \sum_{i=1}^{n}s_{i}\Delta_i$ where $a=x_0\lt x_1\lt\cdots\lt x_n=b$ is the partition, $s_i$ is supremum of the values of $f(x)$ on $[x_{i-1},x_i]$, and $\Delta_i=x_i-x_{i-1}$. Similarly, let $\underline{S}(f,P)$ be the lower Riemann sum of $f$ on $[a,b]$ relative to the partition $P$, that is $\underline{S}(f,P) = \sum_{i=1}^n m_i\Delta_i$ where $m_i$ is the infimum of the values of $f$ on $[x_{i-1},x_i]$.

$f$ is integrable on $[a,b]$ if and only if it is bounded, and for any sequence of partitions $P_n$ of $[a,b]$ such that the mesh size $\lVert P_n\rVert\to 0$ as $n\to\infty$, we have $\lim_{n\to\infty}\underline{S}(f,P_n) = \lim_{n\to\infty}\overline{S}(f,P_n),$ or equivalently, if $\lim_{n\to\infty}\Bigl( \overline{S}(f,P_n) - \underline{S}(f,P_n)\Bigr) = 0.$

Now think about what $\overline{S}(f,P_n) - \underline{S}(f,P_n)$ represents. At any given subinterval of the partition $[x_i,x_{i+1}]$, we are taking $\bigl(s_i - m_i\bigr)\Delta_i$. Now, $\Delta_i$ is the length of the interval; $s_i$ is the supremum of the values the function takes, and $m_i$ is the infimum value the function takes. That means that the graph of the function on this interval is contained in the rectangle $[x_i,x_{i+1}]\times[m_i,s_i]$, which has area exactly $\bigl(s_i-m_i\bigr)\Delta_i$. That is, the difference between $\overline{S}(f,P_n)$ and $\underline{S}(f,P_n)$ can be interpreted as the sum of the areas of a collection of rectangles that contains the graph of $y=f(x)$.

Can you take it from here? (Note that the above does not depend in any way on whether $f$ is positive, negative, chaotic, continuous, or not; just on the fact that it is bounded and integrable).