# Derivatives and Lebesgue integrals in $\mathbb{R}^{2}$

I tried to solve the following question while studying for my exams, and since I have no solution for it, I would love it if you could tell me if my proof makes sense. The context is Lebesgue integrals.

### Question

Let $f:[0,1]\times[0,1] \rightarrow \mathbb{R}$, where:

1. For every $t\in[0,1]$, the function $x \rightarrow f(x,t)$ is integrable on $[0,1]$ (meaning $\int_{0}^{1}f(x,t)dx < \infty$)
2. For every $(x,t) \in [0,1]\times[0,1]$, the partial derivative: $$g(x,t) := \frac{\mathrm{d}f}{\mathrm{d} t}(x,t)$$ exists.
3. And finally: $$M:= sup\left \{|g(x,t)| : 0\leq x,t\leq1 \right \} < \infty$$

Show that $$F(t) := \int_{0}^{1} f(x,t)dx$$ is derivable in [0,1], and that $$F'(t) = \int_{0}^{1}g(x,t)dx$$

### My Solution

Let $$g_n(x,t) := \frac{f(x, t + \frac{1}{n}) - f(x,t)}{\frac{1}{n }}.$$ So according to (2), $\underset{n \rightarrow \infty}{lim} g_n = g(x, t)$, and from (3), there exists some $M'$ such that for large enough $n$, all $g_n$ are bounded by the same $M'$.

Let

$$F'_n(t) := \frac{\int_{0}^{1} f(x,t + \frac{1}{n})dx - \int_{0}^{1} f(x,t)}{\frac{1}{n}}dx$$

So $F'_n(t) = \int_{0}^{1} g_n(t)dx$, and according to the bounded convergence theorem, $\underset{n \rightarrow \infty}{lim} F'_n(t) = \int_{0}^{1}g(x,t)dx$, and since from (3), $f$ is Lipschitz in $[0,1]\times[0,1]$, and therefor g is integrable for every t, I think we are done.

Thanks!

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+1 for showing your work. I think there is a typo in the first line, it should be $\lim_n g_n(x,t)=g(x,t)$. You can express the bound $M'$ with $M$ thanks to the mean value theorem. In fact $g$ is integrable since it's bounded over a finite measure set. Last thing: integrable means that $\int_{[0,1]}|f(x,t)|<\infty$ (you need the absolute value). –  Davide Giraudo Feb 25 '12 at 11:36
one more thing is that you have to work with a general $t_n\to 0$, not just $1/n$. –  Ashok Feb 25 '12 at 11:59
@DavideGiraudo Thanks! In fact we defined integrable functions without the absolute value, and then proved that f is integrable iff |f| is, but since it's the second time I get this comment, I may start using |f| online instead ;). –  Hila Feb 25 '12 at 12:02
@Ashok Now that you say it, it makes sense, but all the proofs in my notebook work with $\frac{1}{n}$, so now I'm confused. Did my professor leave something out? –  Hila Feb 25 '12 at 12:06
@Hila You may use 1/n or any other sequence the important is the $g_n$ are converging to the properly limit. The bound of the $g_n$ cannot come from the mean value since the derivatives derivatives are only almost everywhere (a.e.). I think in exam you should explain better this bound since it is crucial. –  checkmath Feb 25 '12 at 12:33