I'm trying to understand this question:
If $I$, $J$ are intervals on $\mathbb{R}$ and $I \subset J$ show that $L^1(J) \subset L^1(I)$.
edit: I was reading a book called Lebesgue Integration and Measure by Weir, I wasn't too familiar with Lp spaces, thus I was stuck on this question.
In general, if $I,J\subset \mathbb{R}$ and you need to prove that $J\subset I$ so, you show that if $x\in J$, so $x\in I$.
Now, by definition the space $L^{1}(X)$ consists of all real (in this case)-valued measurables functions on $X$ that satisfy $$\int_{X}|f(x)|^{1}dx<+\infty.$$
Let $I:=]a,b[$ and $ J=]c,d[$ such that $I\subset J$. Note that, it's equivalent to $]a,b[\subset ]c,d]$. Taking a real-valued measurable function on $L^{1}(J)$, so you have that $$\int_{J}|f(x)|{\rm d}x=\int_{c}^{d}|f(x)|{\rm d}x<+\infty$$ Now, by additivy you have that $$\int_{c}^{d}|f(x)|{\rm d}x=\int_{c}^{a}|f(x)|{\rm d}x +\int_{a}^{b}|f(x)|{\rm d}x+\int_{b}^{d}|f(x)|{\rm d}x<+\infty.$$ Can you continue from here?