1)Is it true that any function can be decomposed as a difference of its positive and its negative part as $f=f^{+}-f^{-}$ or that function should belong to $\mathcal{L}^{1}(\mathbb{R})$. Also if that function doesn't belong to $\mathcal{L}^{1}(\mathbb{R})$ but belongs to $\mathcal{L}^{2}(\mathbb{R})$ then can we still write the above decomposition.
2)If $\int_\mathbb{R}f(x) dx=0$ then can we say that $f\in\mathcal{L}^{1}(\mathbb{R}).$
1) Just define $f^{+}\left(x\right)=\max\left\{ 0,f\left(x\right)\right\} $ and $f^{-}\left(x\right)=\max\left\{ 0,-f\left(x\right)\right\} $.
Then $f^{+}$ and $f^{-}$ are nonnegative functions with $\left|f\right|=f^{+}+f^{-}$ and $f=f^{+}-f^{-}$.
This is true for any function $f$.
2) $\int f\left(x\right)dx=0$ can only be true iff the integrals $\int f^{+}\left(x\right)dx$ and $\int f^{-}\left(x\right)dx$ are finite and equal.
In that case $\int f\left(x\right)dx:=\int f^{+}\left(x\right)dx-\int f^{-}\left(x\right)dx=0$.
And also $\int\left|f\right|\left(x\right)dx:=\int f^{+}\left(x\right)dx+\int f^{-}\left(x\right)dx<\infty$.
